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Doctoral Thesis

Ramsey and Universality Properties of Random Graphs

Author(s): Nenadov, Rajko

Publication Date: 2016

Permanent Link: https://doi.org/10.3929/ethz-a-010711006

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Diss. ETH No. 23559

Ramsey and UniversalityProperties of Random Graphs

A thesis submitted to attain the degree of

DOCTOR OF SCIENCES of ETH Zurich(Dr. sc. ETH Zurich)

presented by

Rajko NenadovMaster of Science ETH in Computer Science

born on 14.11.1987

citizen of Serbia

accepted on the recommendation of

Prof. Dr. Angelika Steger, examiner

Prof. Dr. Michael Krivelevich, co-examiner

2016

To my Dad, for instilling in mea passion for mathematics

Contents

Abstract iii

Zusammenfassung vii

Acknowledgments xi

1 Introduction 11.1 Ramsey’s theorem . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Ramsey numbers . . . . . . . . . . . . . . . . . . 51.2 Ramsey’s theorem and random graphs . . . . . . . . . . 8

1.2.1 Anti-Ramsey properties of random graphs . . . . 101.2.2 Excursion into positional games . . . . . . . . . . 11

1.3 Size Ramsey numbers . . . . . . . . . . . . . . . . . . . 111.3.1 Universality of random graphs . . . . . . . . . . 13

2 Preliminaries 152.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Graph-theoretic notation . . . . . . . . . . . . . 162.2 Probabilistic tools and estimates . . . . . . . . . . . . . 172.3 Graph density measures and decompositions . . . . . . . 20

3 Two selected proofs 23i

ii Contents

3.1 A short proof of the Rödl-Ruciński Theorem – 1-statement 233.2 Universality for degenerate graphs . . . . . . . . . . . . 27

4 Ramsey-type problems in random graphs 314.1 Results – old and new . . . . . . . . . . . . . . . . . . . 32

4.1.1 Outline of the proof . . . . . . . . . . . . . . . . 364.2 A general framework . . . . . . . . . . . . . . . . . . . . 37

4.2.1 Outline of the method . . . . . . . . . . . . . . . 374.2.2 Proof of the framework theorem . . . . . . . . . 39

4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.1 Anti-Ramsey property – proper colourings . . . . 514.3.2 Anti-Ramsey property – 2-bounded colourings . 574.3.3 Ramsey property for graphs and hypergraph cliques 71

5 Universality of random graphs 735.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.1 Overview of the proof . . . . . . . . . . . . . . . 755.2 Universality for small graphs . . . . . . . . . . . . . . . 765.3 Disjoint representatives in hypergraphs . . . . . . . . . . 805.4 Proof of the main theorem . . . . . . . . . . . . . . . . . 85

6 Size-Ramsey numbers of graphs with bounded degree 896.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1.1 Overview of the proof . . . . . . . . . . . . . . . 916.2 Decomposition of triangle-free graphs . . . . . . . . . . . 936.3 The regularity method . . . . . . . . . . . . . . . . . . . 98

6.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . 996.3.2 Lower regularity . . . . . . . . . . . . . . . . . . 1006.3.3 KŁR conjecture and cycles . . . . . . . . . . . . 104

6.4 Proof of the main theorem . . . . . . . . . . . . . . . . . 107

Bibliography 119

Abstract

This thesis investigates the interplay between two branches of discretemathematics: Ramsey theory and random graphs.

The origins of Ramsey theory can already be found in the work of Hilbertfrom the early 20th century. However, it was a simple but profound re-sult of Frank P. Ramsey from 1930 which marked the beginning of thenew field in mathematics. Phrased in graph theory terms, Ramsey’stheorem states that for every `-uniform hypergraph H and sufficientlylarge complete `-uniform hypergraph K(`)

n , no matter how one coloursthe edges of K(`)

n with two colours there will always exist a monochro-matic copy of H, that is a copy of H with all edges having the samecolour. For short, we say that such K(`)

n is Ramsey for H. Many vari-ations of the Ramsey property have been studied since then. Generallyspeaking, we will use the term Ramsey-type property as an umbrellaterm for any such property involving colourings.

Even though it might appear as a naive statement at first, Ramsey’stheorem gave rise to a whole new field with far-reaching results. Per-haps even more important, many techniques and ideas that are nowconsidered to be standard tools in the arsenal of a mathematician havebeen developed in order to tackle questions in Ramsey theory. Thus itis a fruitful playground whose problems serve as benchmarks for ideas.

iii

iv Abstract

Various questions derived from Ramsey’s theorem continue to challengemathematicians.

The study of random graphs started with the seminal work of Erdős andRényi from 1959. Rather than just being interesting structures on theirown, it turned out that some of the best possible (or best known) con-structions of graphs with certain properties stem from random graphs.This twofold view on random graphs is also present here: in the firstpart of the thesis we study random graphs for their own sake, whilein the second part we use them to construct sparse graphs which areRamsey for a special family of graphs.

We consider the binomial random graph model G(n, p). Given a param-eter p ∈ [0, 1], a random graph G ∼ G(n, p) on n vertices is generated bysimply including each possible edge with probability p, independently.Bollobás and Thomason showed that random graphs exhibit a ratherpeculiar phenomenon: given a monotone increasing graph property P(i.e. a property which is preserved under edge addition) there exists afunction p0 = p0(n) such that if p p0 (resp. p p0) then G(n, p) hasthe property P with probability tending to 1 (resp. 0) as n goes to in-finity. In other words, random graphs exhibit threshold behaviour withrespect to monotone properties. For example, P can be the propertyof being connected, containing a Hamilton cycle, etc. The typical prob-lem in random graph theory is thus to determine such a function for achosen property. We make a small contribution towards understandingthresholds for Ramsey-type properties in random graphs.

In the first part of the thesis we build upon a classic theorem of Rödland Ruciński which determines the threshold for the property of beingRamsey for a graph H. Our first contribution is a short proof of theupper bound on the threshold in the Rödl-Ruciński Theorem using theso-called hypergraph containers method developed by Balogh, Morrisand Samotij and independently by Saxton and Thomason. Taking intoaccount that we use heavy machinery, this is a rather humble contri-bution. Nonetheless, it has both educational and research characters:our proof has appeared in the recent monograph on random graphs byFrieze and Karoński and the ideas of the proof were further utilised byother groups of authors.

Our second contribution is a unifying approach for proving lower boundson thresholds for various Ramsey-type properties. In particular, wedevelop a framework which reduces the problem of showing that G(n, p)does not have a Ramsey-type property to showing that graphs with

Abstract v

certain local density conditions do not have such a property. In otherwords, we reduce the probabilistic question to a deterministic one. Thisapproach is then demonstrated to obtain a lower bound for the Ramseyproperty (reproving the Rödl-Ruciński Theorem), anti-Ramsey propertyfor r-bounded colourings and proper colourings, and the Maker-BreakerH-game.

In the second part of the thesis we study size Ramsey numbers of graphswith constant maximum degree, a class of graphs that has recentlyattracted considerable attention in the theory of random graphs. SizeRamsey number ofH, denoted by r(H), is defined as the smallestm ∈ Nsuch that there exists a graph G with m edges which is Ramsey for H.Given a graph H with maximum degree at most some constant ∆, it isof interest to determine the order of r(H). On the one hand, Rödl andSzemerédi showed that for every sufficiently large n there exists a 3-regular graph H such that r(H) ≥ n logc n for some constant c > 0. Onthe other hand, the result of Rödl, Kohayakawa, Szemerédi and Schachtshows that there exists a constant C > 0 such that

r(H) ≤ Cn2−1/∆ log1/∆ n

for every graph H with n vertices and maximum degree ∆. This leavesa gap between the two bounds. We give a polynomial improvement ofthe upper bound in the case where H is triangle-free. This includes,among others, bipartite graphs. Similarly to Rödl et al. we use randomgraphs to find a witness for r(H). There is evidence that in some casesthe bound we obtain is close to the best possible using random graphs.

It turns out that our proof on the upper bound on r(H) actually impliesa stronger statement. In particular, we show that there exists a graphG with n2−1/∆−α edges, for some α = α(∆) > 0, such that for everycolouring of the edges of G one of the colours contains every graph withat most n vertices and maximum degree at most ∆. In other words, oneof the colours is universal for such a family of graphs. Therefore, as awarm-up, we first consider the question of finding the threshold for theproperty that G(n, p) is universal for graphs with bounded maximumdegree. Improving upon the result of Alon, Capalbo, Kohayakawa, Rödl,Ruciński and Szemerédi, we determine (up to the logarithmic factor) thethreshold for such property when ∆ = 3 and give the best known upperbound in case ∆ > 3. Ideas developed for this problem are then reusedto improve the size Ramsey number.

vi Abstract

Zusammenfassung

Ursprünge der Ramsey-Theorie können bereits in der Arbeit von Hil-bert aus dem frühen 20. Jahrhundert gefunden werden. Es war jedochein einfaches, aber tiefes Ergebnis von Frank P. Ramsey aus dem Jahr1930, das den Beginn des neuen Feldes in der Mathematik markiert. Indie Sprache der Graphentheorie übersetzt besagt der Satz von Ramseydass für jeden `-uniformen Hypergraphen H und hinreichend grossenund vollständigen `-uniformen Hypergraphen K`

n, jede 2-Färbung derKanten von K`

n eine einfarbige Kopie von H enthält. Kurz gesagt K`n ist

Ramsey für H. Viele Variationen der Ramsey-Eigenschaft wurden seit-dem untersucht. Generell werden wir den Begriff Ramsey-Eigenschaftals Oberbegriff für eine solche Eigenschaft jeder Färbung verwenden.

Auch wenn es zunächst als naive Aussage erscheinen mag, enstand mitdem Satz von Ramsey ein ganz neues Feld mit weitreichenden Ergeb-nissen. Von vielleicht noch grösserer Bedeutung sind viele Technikenund Ideen, die nun Standard-Methoden im Arsenal eines Mathemati-kers sind, und die im Zusammenhang mit Problemen aus der Ramsey-Theorie entwickelt wurden. Das heisst Ramsey-Theorie ist ein Spiel-platz, dessen Probleme als Massstab für neue Ideen dienen. Verschie-dene Fragen, die aus dem Satz von Ramsey abgeleitet sind, fordernMathematiker weiterhin heraus.

vii

viii Zusammenfassung

Die Untersuchung von Zufallsgraphen begann mit der bahnbrechendenArbeit von Erdős und Rényi von 1959. Zufallsgraphen sind nicht nurintrinsisch interessant. Es stellte sich heraus, dass einige der bestenKonstruktionen von Graphen mit bestimmten Eigenschaften von Zu-fallsgraphen stammen. Diese doppelte Ansicht auf Zufallsgraphen wirdauch hier präsentiert: Im ersten Teil der Arbeit untersuchen wir Zufalls-graphen um ihrer selbst willen, während wir sie im zweiten Teil benut-zen, um dünne Graphen zu konstruieren, die Ramsey für eine gewisseFamilie von Graphen sind.

Wir betrachten das binomische Zufallsgraphen-Modell. Für einen Pa-rameter p ∈ [0, 1] wird ein Zufallsgraph G ∼ G(n, p) auf n Knotenerzeugt, indem jede mögliche Kante mit Wahrscheinlichkeit p existiert,wobei alle Kanten unabhängig gewählt werden. Bollobás und Thomasonzeigten, dass Zufallsgraphen sehr spezielle Erscheinung besitzen: Es seiP eine monoton steigende Grapheneigenschaft, das heisst eine Eigen-schaft, die unter Hinzufügen von Kanten erhalten bleibt. Bollobás undThomason zeigten, dass dann eine Funktion p0 = p0(n) existiert so dassG(n, p) die Eigenschaft P mit einer Wahrscheinlichkeit besitzt, die fürp p0 (p p0) gegen 1 (0) geht, wenn n → ∞. Mit anderen Worten,Zufallsgraphen zeigen ein Schwellenverhalten in Bezug auf monotoneEigenschaften. Zum Beispiel kann P die Eigenschaft sein zusammen-hängend zu sein; einen Hamiltonkreis zu enthalten, usw. Das typischeProblem in der Zufallsgraphentheorie ist für eine gegebene Eigenschafteine solche Funktion zu bestimmen. Wir machen einen kleinen Beitragzum Verständnis einiger Schwellenwerte für Ramsey-Eigenschaften inZufallsgraphen.

Im ersten Teil der Arbeit betrachten wir einen klassischen Satz von Rödlund Ruciński, der den Schwellenwert für die Eigenschaft Ramsey füreinen gegebenen Graph H bestimmt. Unser erster Beitrag ist ein kurzerBeweis für die obere Schranke an den Schwellenwert im Rödl-RucińskiSatz mit Hilfe der so genannten Hypergraph-Container-Methode, ent-wickelt von Balogh, Morris und Samotij und unabhängig von Saxtonund Thomason. Auch wenn wir hier schwere Maschinerie verwenden, sohat dies sowohl Bildungs- als auch Forschungswert: Unser Beweis ist inder jüngsten Monographie über Zufallsgraphen von Frieze und Karoń-ski erschienen und die Ideen des Beweises wurden weiter durch andereGruppen von Autoren genutzt.

Unser zweiter Beitrag ist ein verbindender Ansatz für den Nachweisder unteren Grenzen für Schwellenwerte für verschiedene Ramsey Ei-

Zusammenfassung ix

genschaften. Insbesondere entwickeln wir einen Rahmen, der das Pro-blem, zu zeigen dass G(n, p) keine Ramsey-Eigenschaft hat, auf das Pro-blem reduziert zu zeigen, dass Graphen mit bestimmten lokalen Dichte-Bedingungen nicht eine solche Eigenschaft haben. Mit anderen Worten,wir reduzieren die probabilistische Frage zu einer deterministischen. Die-ser Ansatz wird dann genutzt, um verschiedene weitere Resultate zuerhalten.

Im zweiten Teil der Arbeit untersuchen wir size-Ramsey-Zahlen vonGraphen mit konstantem Maximalgrad, eine Klasse von Graphen, dievor kurzem grosse Beachtung in der Theorie der Zufallsgraphen fand.Die size-Ramsey-Zahl von H bezeichnet durch r(H), ist die kleinste na-türliche Zahlm ∈ N, so dass es einen Graphen G mitm Kanten gibt, derRamsey fürH ist. SeiH ein Graph mit Maximalgrad höchstens ∆, wobei∆ eine Konstante ist, so wollen wir die Grösse von r(H) bestimmen. Aufder einen Seite zeigten Rödl und Szemerédi, dass für jedes hinreichendgrosse n ein 3-regulärer-Graph H existiert, so dass r(H) ≥ n logc n füreine Konstante c > 0. Auf der anderen Seite zeigten Rödl, Kohayakawa,Szemerédi und Schacht, dass es eine Konstante C > 0 gibt, so dass

r(H) ≤ Cn2−1/∆ log1/∆ n

für jeden GraphenH mit nKnoten und Maximalgrad ∆. Dies hinterlässteine beträchtliche Lücke. Wir geben eine polynomielle Verbesserung deroberen Schranke an, für den Fall, dass der Graph H dreiecksfrei ist. Diesdeckt unter anderem den Fall der bipartiten Graphen ab. Ähnlich wiebei Rödl et al. verwenden wir Zufallsgraphen um einen Zeugen für r(H)zu finden. Es ist erwiesen, dass in einigen Fällen die Schranke, die wirerhalten, in der Nähe der bestmöglichen Verwendung von Zufallsgraphenist.

Es stellt sich heraus, dass unser Beweis an der oberen Grenze für r(H)tatsächlich eine starke Aussage impliziert. Insbesondere zeigen wir, dassein Graph G existiert mit n2−1/∆−α Kanten, für ein α = α(∆) > 0, sodass für jede Färbung der Kanten von G eine der Farben jeden Graphenmit höchstens n Knoten und Maximalgrad höchstens ∆ enthält. Mitanderen Worten: eine der Farben ist universal für eine solche Familievon Graphen.

Als nächstes betrachten wir daher das Problem, die Schwellenwerte fürdie Eigenschaft zu finden, dass G(n, p) universal für Graphen mit be-schränktem Maximalgrad ist. Wir verbessern ein Ergebnis von Alon,Capalbo, Kohayakawa, Rödl, Ruciński und Szemerédi, indem wir den

x Zusammenfassung

Schwellenwert für eine solche Eigenschaft bestimmen (bis auf einen lo-garithmischen Faktor) für ∆ = 3 und geben die beste bekannte obereSchranke im Fall ∆ > 3. Ideen aus diesem Beweis werden dann verwen-det, um die size-Ramsey-Zahl zu verbessern.

Acknowledgments

After almost six years, first as a master’s and then a doctoral student,my journey at ETH Zurich has come to an end. I would like to ac-knowledge those people without whom this time would not have beenthe fulfilling experience that it was.

I am deeply grateful to my advisor Angelika Steger, for everything shehas done for me from the day I started my masters studies in 2010until the day I moved to Melbourne a couple of months ago. It was herinspiring teaching of the ‘Random graphs and the probabilistic method’course, a classic one at ETH, and challenging research and master’sthesis projects that drew me to the wonderful world of random graphs.Thank you, Angelika, for all the research retreats you organised, all theconferences I have attended thanks to the work we did together and,most importantly, all the hours you spent in front of the blackboardwith me sharing your knowledge! A student could not ask more from asupervisor.

I cannot thank Asaf Ferber enough for being my (unofficial) co-mentor.Thank you for all the ideas you shared, for the patience and the goodtimes we had. We were a good team and I look forward to our manynew collaborations.

Heartfelt thanks to Nemanja Škorić for sharing the passion for com-

xi

xii Acknowledgments

puter science and mathematics since high school. Having you there as asounding board made the research so much fun. If I would start writingall the things for which I am grateful to you, this section would be fartoo long.

I want to thank Michael Krivelevich for taking time from his new du-ties as a dean and co-refereeing my thesis. Your opinion on the workpresented in this thesis is very valuable to me.

I am grateful for the opportunity to collaborate with others not previ-ously mentioned: David Conlon, Daniel Korándi, Frank Mousset, An-dreas Noever, Yury Person, Pascal Pfister, Ueli Peter, Miloš Stojakovićand Benny Sudakov.

Thanks to all past and current members of Angelika’s group, especiallythose whose frequent visits to my office made it a more lively place.

Many thanks to Maja Harris for proofreading (and translating) the ab-stract and introduction and for making my life easier while I was workingon this thesis.

Finally, I want to thank my Mum, Dad and sister for all their love andsupport. Being abroad is easy when you have a family like that at home!

Chapter 1Introduction

Ramsey theory is a branch of mathematics deemed to belong to puremathematics by some and applied mathematics by others. Due to thecombinatorial nature of proofs and challenges concerned with construc-tions of certain examples, it attracts more and more researchers workingin the field of theoretical computer science. As a result of such differentmotivations and backgrounds of researchers, it has seen great develop-ment in the past 75 years and has contributed many tools and techniquesthat have helped shape other fields of mathematics. In short, a typicalRamsey-type question can be stated as follows:

Given a colouring of the elements of some structure with a finitenumber of colours, what kind of monochromatic substructure is

guaranteed to exist?1

2 Chapter 1. Introduction

Working on such innocent-looking questions has led to many deep in-sights and theorems. Probably the most notable ones are the far-reaching generalisations of Van der Waerden’s theorem (which will bestated shortly), namely the Szemerédi’s theorem [Sze75] on arithmeticprogressions in dense subsets of integers and the Green-Tao theorem[GT08] on the arithmetic progressions in prime numbers.

The beginning of what is today known as Ramsey theory dates backto the 1890s and the work of Hilbert [Hil92] on the irreducibility ofrational functions. Needless to say, Hilbert’s interest was not in thecombinatorial nature of the obtained auxiliary colouring result and assuch it did not attract the attention of other mathematicians of thetime.

The next result, similar in spirit, was obtained by Schur [Sch12] some20 years later. It is a beautiful result which gives a flavour of what atypical question in Ramsey theory looks like: given a colouring of natu-ral numbers with a finite number of colours, there exists x, y, z ∈ N suchthat x+y = z and all three numbers have the same colour. Similarly toHilbert, Schur’s motivation did not lie in its combinatorial aspects butagain served as an auxiliary tool for proving a more “serious” numbertheoretic result. However, he helped the development of the field byraising a similar question on the existence of monochromatic arithmeticprogressions: does every colouring of natural numbers with a finite num-ber of colours contain an arbitrarily long arithmetic progression with allelements having the same colour? The same conjecture was proposedindependently by Henry Baudet. For an interesting story and a discus-sion on the origin of the conjecture, refer to [Soi08]. This conjecturewas finally resolved by Van der Waerden [VdW27] in 1927.

An observant reader will notice that the theorems of Schur and Vander Waerden are essentially questions about monochromatic solutionsof a system of linear equations. Indeed, Schur’s theorem shows thatthe equation x + y = z has a monochromatic solution and Van derWaerden’s theorem shows the same for the system of equations of theform xi+1 = d + xi for i ∈ 1, . . . , k − 1 and any k ∈ N. This line ofresearch culminated with the work of Rado [Rad33b, Rad33a, Rad43]who unified all previous results and fully characterised systems of linearequations which contain a monochromatic solution under any colouring.Since then, the research has moved onto other structures including hy-pergraphs, posets, parameter sets and boolean lattices, to name a few,reaching further milestones such as the Hales-Jewett theorem [HJ63].

3

For a detailed account on the historically important achievements inRamsey theory of discrete structures, refer to the recent monograph byPrömel [Prö13].

In the following section we introduce Ramsey’s theorem, a result fromwhich the whole field derives its name. Most of the work in this thesisis influenced by Ramsey’s theorem and aimed towards understandingits consequences and underlying concepts. We also feel obliged to usethe opportunity to introduce the notion of Ramsey numbers, one of thedriving forces of the field. Even though not directly related to results ofthe thesis, it will serve to point out some highlights and achievementsof graph Ramsey theory and introduce questions similar to the oneswe consider later. In Section 1.2 we introduce Ramsey’s theorem forrandom graphs, proved by Rödl and Ruciński, around which the firstpart of the thesis is built on. In Section 1.3 we introduce size Ramseynumbers, the main topic of the second part of the thesis.

Thesis overview

Since the goal of every thesis is to deepen the knowledge of a field, the-orems and proofs are likely to be technical and interesting mostly topeople working on similar topics. To lessen this problem to some ex-tent, in Chapter 3 we selected two of our proofs which give a flavour ofthe thesis and at the same time are self-contained and short enough tobe accessible to mathematicians working in discrete mathematics andother fields. The first proof from this chapter has appeared in the recentmonograph on random graphs by Frieze and Karoński [FK15]. Chap-ter 4 studies various extensions of the Rödl-Ruciński Theorem on theconnection between Ramsey’s theorem and random graphs. In partic-ular, we present a unifying approach for proving most of the knownresults and apply it further to derive some new ones. In Chapter 5 wediverge from Ramsey theory and study universality of random graphswith respect to the family of large graphs with bounded maximum de-gree. Ideas from this chapter are subsequently used in Chapter 6 toimprove bounds on size Ramsey numbers of triangle-free graphs withbounded maximum degree. As a special case, this improves size Ramseynumbers of bipartite graphs.


1.1 Ramsey’s theorem

Around the same time that Van der Waerden proved the Schur-Bauderconjecture, a different result with a similar flavour was obtained byRamsey [Ram30]. Working on propositional logic, he proved a simplebut profound statement (Ramsey’s theorem), after which the entire areawas later named. Even now, 85 years later, certain aspects of thistheorem are far from fully understood and continue to challenge newgenerations of mathematicians. We state it using the modern notation:given hypergraphs G and H and a natural number c ∈ N, we say thatG is Ramsey for H (c-Ramsey, if we want to be precise), or

G→ (H)c

for short, if for every colouring of the edges of G with c colours thereexists a copy ofH inG with all edges having the same colour. We refer tosuch a copy as monochromatic. The arrow notation is usually attributedto Erdős and Rado [ER53]. Moreover, we say that a hypergraph H is`-uniform (`-graph for short) if all its edges have size `.

Theorem 1.1 (Ramsey’s theorem). Given `, c ∈ N and `-graph H,there exists N ∈ N such that

K(`)N → (H)c,

that is, every c-colouring of the edges of the complete `-graph with Nvertices contains a monochromatic copy of H.

Note that Ramsey’s original statement was for the case where H is aninfinite complete `-graph. Using plain language, he proved that for anycolouring of `-subsets of N with a finite number of colours there existsan infinite subset H ⊆ N such that every `-subset of H has the samecolour. Theorem 1.1 can be obtained from this result using König’slemma and by replacing an arbitrary `-graph H with a complete onewith the same number vertices.

It is fair to say that the popularity of Ramsey’s theorem owes a greatdeal to the work of Erdős and Szekeres [ES35]. They used it to provea result in combinatorial geometry which might seem unrelated at first:for every n ∈ N and every sufficiently large set of points in the plane ingeneral position there exists a subset of exactly n points which form thevertices of a convex polygon (see [Joh86] for a short proof). Since one

1.1. Ramsey’s theorem 5

way to look at Ramsey’s theorem is as a generalisation of the pigeon-hole principle, applications such as the one of Erdős and Szekeres arenot entirely surprising (bearing in mind applications of the pigeonholeprinciple) though are very interesting and intriguing.

1.1.1 Ramsey numbers

Ramsey’s theorem immediately brings attention to the parameterN , thenumber of elements needed for the Ramsey property to hold. Indeed,the following question has been one of the driving forces of the wholefield:

What is the smallest value of N such that K(`)N → (H)c?

This number is usually referred to as the Ramsey number and denotedby r(H, c). As already mentioned, Ramsey’s original statement was interms of infinite sets and, consequently, the finite version obtained as acorollary does not give any quantitative bounds on r(H, c). Erdős andSzekeres [ES35] gave a constructive proof of Ramsey’s theorem whichprovides a recursive formula for an upper bound on r(K

(`)n , c). This

implicitly gives a bound for an arbitrary `-graph H with n vertices asit is easy to see that r(H, c) ≤ r(K(k)

n , c). We will see shortly that suchan upper bound can be very crude as there exist rich families of graphswith n vertices for which the Ramsey number is exponentially smallerthan r(Kn, c). In particular, there are cases where the Erdős-Szekeresbound can be significantly improved and cases for which even a minorimprovement is notoriously hard. We start with the latter.

For the case where we have two colours and H = Kn is a completegraph, the Erdős-Szekeres bound has a particularly nice closed form,

r(Kn, 2) ≤(

2n− 2

n− 1

).

This is a classic bound in Ramsey theory and even though it is based ona simple induction it turns out to be surprisingly difficult to improve.After a series of results showing that r(Kn, 2) is asymptotically smallerthan

(2n−2n−1

), the first superpolynomial improvement was obtained only

recently by Conlon [Con09b]. On the other hand, Erdős [Erd47] showedthat r(Kn, 2) is indeed exponential. This is a historically important


proof as it is considered to be one of the first applications of the prob-abilistic method in combinatorics. Similarly as in the case of the upperbound, improving the lower bound has turned out to be notoriously dif-ficult and after almost 70 years the only progress was made by Spencer[Spe75]. Even though it improved Erdős’ bound by a factor of only2, it is significant as one the early applications of the Lovász LocalLemma [EL75], an important tool in modern combinatorics. However,these results leave a significant gap between the best known upper andthe lower bound on r(Kn, 2). Determining the order of magnitude ofr(Kn, 2) remains one of the biggest open problems in Ramsey theory.

Another interesting line of research related to the case H = Kn is toderandomise the lower bound of Erdős. As previously mentioned, thisbound is based on a random colouring: for each edge choose (indepen-dently) one of the colours uniformly at random. A simple calculationshows that with positive probability this produces a colouring of a com-plete graph on at most 2n/2 vertices without a monochromatic Kn, thussuch a colouring exists. An ingenious construction of Frankl and Wilson[FW81] gives an explicit colouring of a complete graph on nc logn/ log logn

vertices with the same property, for some constant c > 0. Very recently,Barak, Rao, Shaltiel and Wigderson [BRSW12] gave a construction ofthe colouring which improves this to

22(log logn)1+ε

.

However, finding an explicit colouring even for a complete graph on(1+ε)n vertices without a monochromatic Kn seems to be very difficult.

After the complete graphs, the next classical topic in Ramsey theoryare graphs which have only “few” edges. The starting point is a workof Gerencsér and Gyárfás [GG67] where they determined the 2-colourRamsey number of Pn, the path on n vertices,

r(Pn, 2) = b(3n− 2)/2c.

Not only is r(Pn, 2) linear in n, in contrast to r(Kn, 2), it is also oneof the rare examples where the exact value is known. To illustrate thedifficulty of the multi-colour case (i.e. c > 2), note that the exact resultfor r(Pn, 3) was obtained only recently, some 40 years later, by Gyárfás,Ruszinkó, Sárközy and Szemerédi [GRSS07].

The study of more general sparse graphs was initiated by Burr andErdős [BE75] where they put forward two conjectures which we state

1.1. Ramsey’s theorem 7

shortly. To simplify notation, for the rest of the thesis we assume thatthere are only two colours (unless stated otherwise) and write r(H) todenote r(H, 2). Moreover, we write G→ H to denote G→ (H)2.

Conjecture 1.2. For every ∆ ≥ 2 there exists a constant C > 0 suchthat if H is a graph with n vertices and maximum degree at most ∆ thenr(H) ≤ Cn.

This conjecture was resolved in the affirmative by Chvátal, Rödl, Sze-merédi and Trotter [CRST83] as one of the early successes of Szemerédi’sRegularity Lemma. However, the dependency of the resulting constantC on ∆ was quite weak as is often the case in applications of the Reg-ularity Lemma. Finding alternative proofs which give a better depen-dency has received considerable attention ever since and, as a result,asymptotically optimal bounds are now known for some special casessuch as where H is a bipartite graph (see [Con09a, FS09]). For thegeneral case the best known bound C(∆) ≤ 2O(∆ log ∆) is due to Con-lon, Fox and Sudakov [CFS12]. On the other hand, Graham, Rödl andRuciński [GRR01] showed that there exists a constant c′ > 0 such thatC(∆) ≥ 2c

′∆. It would be of great interest to determine the correctorder of magnitude of C(∆).

Maybe even more importantly than the result itself, improving the de-pendency of C has stimulated development of techniques such as thedependent random choice. This has very recently played a major role inthe proof of the second conjecture of Erdős and Burr.

Conjecture 1.3. For every d ≥ 2 there exists a constant C > 0 suchthat if H is a d-degenerate graph with n vertices then r(H) ≤ Cn.

Recall that a graph H is d-degenerate if every induced subgraph of Hhas a vertex of degree at most d. Equivalently, it is d-degenerate if thereexists an ordering of the vertices, say v1, . . . , vn, such that each vertexhas at most d neighbours to the “left”, i.e., vi has at most d neighboursin v1, . . . , vi−1. Conjecture 1.3 was recently proved by Lee [Lee15].

Families of bounded-degree graphs (i.e. graphs with maximum degreeat most some constant ∆) and d-degenerate graphs play an importantrole in the second part of this thesis. In particular, in Chapter 6 westudy a question similar to Conjecture 1.2 but with respect to a differentparameter, namely the size Ramsey number. Moreover, ideas for provingthe main result of Chapter 6 are based on proofs from Chapter 3 and


Chapter 5 which deal with universality for families of d-degenerate andbounded-degree graphs.

So far we have illustrated some of the highlights and challenges of Ram-sey theory and, in particular, graph Ramsey theory. For a thoroughtreatment and overview of graph Ramsey theory, refer to a classic mono-graph by Graham, Rothschild and Spencer [GRS90] and a recent surveyarticle by Conlon, Fox and Sudakov [CFS15].

In the remaining sections of this chapter we turn our attention to theproblems studied in this thesis.

1.2 Ramsey’s theorem and random graphs

A priori it is not clear whether Ramsey’s theorem follows from thedensity of complete hypergraphs or the rich underlying structure. Ifthe latter is the case then is it possible to replace K(k)

N in Ramsey’stheorem by a sparser hypergraph G which preserves such a structure?The evidence that this is the case comes from the work of Folkman[Fol70] who proved that for every complete graph on n vertices thereexists a graph G such that G→ Kn and G does not contain a completegraph Kn+1 as a subgraph. This was extended to more colours byNešetřil and Rödl [NR76].

Nowadays, the easiest way to prove such a result is by studying Ramseyproperties of random (hyper)graphs. In this thesis we deal with thebinomial random `-graph model G(`)(n, p). Given parameters n ∈ Nand p ∈ [0, 1], the random `-graph G(`)(n, p) is defined as follows: thevertex set V is of size n and each of the

(V`

)possible edges among

` vertices in V is present with probability p independently. In otherwords, G(`)(n, p) is a probability distribution over all `-graphs with nvertices where the probability of obtaining a particular `-graph H withm edges is

pm(1− p)(n`)−m.

One of the intriguing aspects of random `-graphs is that they exhibit thethreshold behaviour for monotone increasing properties (i.e. propertieswhich are preserved under edge addition). That is, for every monotoneincreasing property P (e.g. P contains all connected graphs, all graphswhich are Ramsey for some H, etc.) there exists a function p0 = p0(n)

1.2. Ramsey’s theorem and random graphs 9

such that

limn→∞

Pr[G(`)(n, p) ∈ P] =

1, p p0

0, p p0.

This was proved by Bollobás and Thomason [BT87]. For an introductionto random graph theory, refer to classic monographs by Bollobás [Bol98]and Janson, Ruciński and Łuczak [JŁR11] and a recent one by Friezeand Karoński [FK15].

The study of Ramsey properties in random graphs was initiated byŁuczak, Ruciński and Voigt [ŁRV92] who studied the vertex-colouringcase and determined the threshold for the property G(n, p) → K3. Ina series of papers, Rödl and Ruciński [RR93, RR94, RR95] then de-termined the threshold for G(n, p) → (H)c for every graph H and anarbitrary number of colours. Progress on the corresponding problem forhypergraphs was made recently by Friedgut, Rödl and Schacht [FRS10]and independently by Conlon and Gowers [CG10] who obtained anupper bound analogous to the graph case. However, the question ofwhether there exists a matching lower bound remains open with somepartial results obtained in [Tho13].

In the first part of Chapter 3 we give a short proof of the so-called 1-statement (i.e. the case p p0) of the Rödl-Ruciński Theorem, whichappeared in [NS16]. For simplicity we limit ourselves to the graph case,although the same proof yields the hypergraph version as well. The ideasfrom [NS16] have been subsequently used by Rödl, Ruciński and Schacht[RRS16] to obtain new bounds on the so-called Folkman number, avariation of the Ramsey number inspired by the previously mentionedtheorems of Folkman and Nešetřil and Rödl.

The corresponding 0-statements (i.e. the case p p0) are considered inChapter 4. We start by developing a framework which gives a unified ap-proach to previously studied Ramsey-type problems in random graphs.In Section 4.3 we use it to derive a short proof of the 0-statement ofthe Rödl-Ruciński Theorem and further determine thresholds for vari-ous other Ramsey-type properties. Most of the content of Chapter 4 isbased on [NŠS15a, NSS15b, NS16] and the as yet unpublished [NPŠS14].

Next, we mention some of the variations of the Ramsey property.


1.2.1 Anti-Ramsey properties of random graphs

If we allow colourings with an unbounded number of colours we arrive atthe so-called anti-Ramsey problem. Instead of looking for a monochro-matic copy of a hypergraph H, we are interested in finding a rainbowcopy, i.e., a copy of H in which each edge has a different colour. Toavoid trivialities one needs to forbid colourings with too few colours.This has been done in several different ways. In this thesis we considerthe following two restrictions: (i) we bound the number of times eachcolour is used or (ii) require that the colouring is proper (i.e. no twoincident edges have the same colour).

The case where each colour is used at most r times (an r-bounded colour-ing) was first considered by Lefmann, Rödl and Wysocka [LRW96].They studied the question of determining the largest ` = `(n) such thatany r-bounded colouring of a complete graph Kn contains a rainbowcopy of K`. Bohman, Frieze, Pikhurko and Smyth [BFPS10] initiatedthe study of a similar question in G(n, p). They proved that given agraph H and a constant r ≥ r(H), the threshold for the r-boundedanti-Ramsey property matches (asymptotically) the threshold for thecorresponding Ramsey property. We improve their result by showingthat the same holds for most cases even if r = 2 and also extend itto complete hypergraphs. This is optimal as for r = 1 the problembecomes trivial (any copy of H is by the definition rainbow).

The first result on the relation between random graphs and the proper-colouring version of the anti-Ramsey property comes from the followingquestion raised by Spencer: is it true that for every g there exists agraph G with girth at least g such that every proper colouring of Gcontains a rainbow copy of C`, for some `? The question was answeredin the affirmative by Rödl and Tuza [RT92] who proved that for every` there exists sufficiently small p = p(n) such that G(n, p) has sucha property. Recently, Kohayakawa, Kostadinidis and Mota [KKM11,KKM14a] started a systematic study of this property in random graphs.In particular, they proved that the upper bound matches that of theRamsey property. To further demonstrate our method, we prove thematching lower bound for sufficiently large complete graphs and cycles.

1.3. Size Ramsey numbers 11

1.2.2 Excursion into positional games

Combinatorial games are games like Tic-Tac-Toe or Chess in which eachplayer has perfect information and players move sequentially. Outcomesof such games can thus, at least in principle, be predicted by enumer-ating all possible ways in which the game may evolve. But, of course,such complete enumerations usually exceed available computing powers,which keeps these games interesting to study.

Here we take a look at the Maker-Breaker H-game. Given a (large)graph G and a (small) graph H, the H-game on G is played on theedges of G and the winning sets are the edge sets of all copies of Happearing in G as subgraphs. Maker and Breaker alternately claimunclaimed edges of the graph G until all the edges are claimed. Makerwins if he claims all the edges of a copy of H in which case we say thatthe graph G is Maker’s win. Otherwise Breaker wins, in which case wesay that the graph G is Breaker’s win.

The study of the H-game where G is a random graph was initiated byStojaković and Szabó [SS05], giving approximate bounds on the thresh-old for the property that Maker has a winning strategy (Maker propertyfor short) in case H is a complete graph. The threshold for this casewas later determined by Müller and Stojaković [MS14]. We extend theirresult to all graphs which satisfy certain mild conditions. This includes,among others, all graphs which are either K3-free or contain a subgraphwhich is denser than K3. Building upon our methods, Dean and Kriv-elevich [DK16] have recently extended these results further to a differenttype of games, namely the Waiter-Client games, and also generalisedour result to the biased case.

1.3 Size Ramsey numbers

The notion of size Ramsey numbers was introduced by Erdős, Faudree,Rousseau and Schelp [EFRS78]. Given a graph H, the size Ramseynumber r(H) is the smallest integer m such that there exists a graph Gwith m edges which is Ramsey for H.

Note that the Ramsey number immediately implies an upper boundr(H) ≤

(r(H)

2

). Surprisingly, this is tight in case H = Kn, as observed

by Chvátal. This can be seen as follows: If a graph G has less than(r(Kn)

2

)edges then the chromatic number of G is strictly less than r(Kn).


Let V1, . . . , Vr(Kn)−1 be a partition of G into r(Kn) − 1 independentsets and consider a two-colouring of the edges of Kr(Kn)−1 without amonochromatic copy of Kn. Colouring every edge between Vi and Vj inG with the colour of the edge ij ∈ E(Kr(Kn)−1) avoids monochromaticcopies of Kn in G.

The famous result of Beck [Bec83] shows that the trivial bound r(H) ≤(r(H)

2

)can be significantly improved in the case where H is a path

on n vertices. In particular, he showed that there exists a constantC > 0 such that r(Pn) ≤ Cn. Moreover, he raised a question in [Bec90]of whether r(H) grows linearly for bounded-degree graphs. This wasproven for trees by Friedman and Pippenger [FP87] and cycles by Hax-ell, Kohayakawa and Łuczak [HKŁ95]. However, the general case wasanswered in the negative by Rödl and Szemerédi [RS00], who showedthat for every sufficiently large n there exists a graph H with n verticesand maximum degree 3 such that r(H) ≥ n logc n, for some constantc > 0. In the same paper they conjectured that logc n can be improvedto nε for some constant ε > 0.

The best known upper bound for size Ramsey numbers of graphs withmaximum degree ∆ is due to Kohayakawa, Rödl, Schacht and Szemerédi[KRSS11],

r(H) ≤ Cn2−1/∆(log n)1/∆, (1.1)which is far from the lower bound of Rödl and Szemerédi. In Chapter6 we obtain a polynomial improvement over (1.1) in the case whereH is triangle-free (i.e. it does not contain K3 as a subgraph). Thisincludes, among others, bipartite graphs. As the bound we obtain isonly marginally better than (1.1), it is instructive to explain where itcomes from and why is it challenging to improve.

Similarly as in [KRSS11], we use G(N, p) with N = O(n) to show theexistence of a Ramsey graph G with the desired number of edges. Notethat (log n/n)1/∆ is roughly the lowest value of p at which we can ex-pect that every collection of ∆ vertices in G(N, p) has many neigh-bours in common, a property which is extremely useful if one wishesto embed a graph H of maximum degree ∆. In particular, it allowsone to embed such H iteratively vertex-by-vertex (or, more specifi-cally, a batch of independent vertices at a time), a strategy whichhas been used successfully in various random-like scenarios (e.g. see[ABH+13, ABET15, AF92, DKRR15, KRSS11]). Since the number ofedges of G(N, p) is concentrated around N2p n2p, this gives thenumber of edges as in (1.1). However, if p is asymptotically below

1.3. Size Ramsey numbers 13

(log n/n)1/∆ a new embedding scheme is required. Therefore, ratherthan in the result itself (which neither closes the gap nor covers allbounded-degree graphs) our main contribution is an embedding strat-egy which demonstrates that the “natural” bound of n2−1/∆ can be over-come. Moreover, there is evidence that, in some cases, random graphscannot give an improvement much better than the one we obtain.

1.3.1 Universality of random graphs

Instead of asking for a monochromatic copy of a single graph H, westrengthen the Ramsey property by requiring a monochromatic copy ofevery graph from some family H, in the same colour. More precisely, wesay that a graph G is H-universal if it contains a copy of every graphH ∈ H and H-Ramsey-universal if every two-colouring of the edges of Gcontains a monochromatic H-universal graph. Note that the notion ofRamsey-universality corresponds to the notion of partition-universalityfrom [KRSS11]. As it turns out, our proof that G(n, p) is Ramsey forsome triangle-free graph H with bounded degree shows as a by-productthat it is Ramsey-universal for the family of all such graphs. The sametype of result was obtained in [KRSS11]. Therefore, as a first step westudy the universality of random graphs, a problem interesting on itsown. This is done in Chapter 5, which is based on [CFNŠ16]. We remarkthat the proof given here is somewhat simpler and gives a better boundthan the one from [CFNŠ16].

The first result on universality of random graphs was proved by Alon,Capalbo, Kohayakawa, Rödl, Ruciński and Szemerédi [ACK+00], whoshowed that for any ε > 0 and sufficiently large constant C > 0, ifp ≥ (C log n/n)1/∆ the random graph G(n, p) almost surely containsevery graph with (1−ε)n vertices and maximum degree at most ∆. Forthe reasons mentioned earlier it was of interest to find a proof whichovercomes this bound. We achieve this and along the way determinethe threshold (up to the logarithmic factor) of the universality propertyfor ∆ = 3.

In the second part of Chapter 3 we give a simple proof of the relatedproblem. We determine the threshold (again, up to the logarithmicfactor) of the universality property for the family of all d-degeneratebounded-degree graphs. This result is used in the proof of the mainresult in Chapter 5 and the ideas of the proof are used in Chapter 6.


Chapter 2Preliminaries

In this chapter we introduce the basic notation which will be usedthroughout this thesis.

2.1 Basic notation

Whenever in a statement we do not explicitly specify the dependencyof a parameter on previously defined ones, we implicitly assume it de-pends on all of them. Moreover if, for example, a parameter a dependson parameters b, c, d in Lemma 5.2, then throughout the thesis we usea5.2(b, c, d) to denote such dependency (i.e. a5.2 is a function of threevariables in this case).

We use log n for the base two logarithm and lnn for the natural loga-15

16 Chapter 2. Preliminaries

rithm. For an integer k ∈ N, we use [k] to denote the set 1, . . . , k.We use the standard Landau symbols O,Ω,Θ, o, ω to denote the asymp-totic behavior of functions. If a hidden constant depends on some otherconstant ε, we write Ωε(·). We write a b and a b to expressthat a = o(b) and a = ω(a), respectively. We interchangeably use withhigh probability (abbreviated by w.h.p) and asymptotically almost surely(abbreviated by a.a.s) to denote that an event holds with probability1− o(1) as n→∞.

Given reals x, y, ε > 0, we write x = (1 ± ε)y to denote x ∈ ((1 −ε)y, (1 + ε)y). Similarly, we write x 6= (1 ± ε)y if either x ≤ (1 − ε)yor x ≥ (1 + ε)y. To improve readability, we omit floors and ceilingswhenever they are not crucial.

Given a set S, we say that a family of subsets S1, . . . , St if S is apartition of S if Si ∩ Sj = ∅ for distinct i, j ∈ [t] and S = S1 ∪ . . . ∪ St.Note that we allow sets Si to be empty.

2.1.1 Graph-theoretic notation

Our graph-theoretic notation is standard and follows that of [Wes01].We say that a hypergraph H is `-uniform (`-graph for short) if all itsedges have size `. We usually denote a graph by G and we use thefollowing notation for a graph G, sets of vertices U,W , a vertex v andan integer k.

Symbol Definition

V (G) vertex-set of GE(G) edge-set of Gv(G), vG number of vertices of G, v(G) = |V (G)|e(G), eG number of edges of G, e(G) = |E(G)|NG(v) neighborhood of a vertex v ∈ V (G) in G,

NG(v) = u ∈ V (G) : u, v ∈ E(G)

degG(v) degree of a vertex v in G, degG(v) = |NG(v)|δ(G) minimum degree of G, δ(G) = minv∈V (G) degG(v)

2.2. Probabilistic tools and estimates 17

Symbol Definition

∆(G) maximum degree of G, ∆(G) = maxv∈V (G) degG(v)

EG(U) all the edges of G with both endpoints in U

EG(U) = u,w ∈ E(G) : u, v ∈ U

EG(U,W ) all the edges of G with one endpoint in U and W

EG(U,W ) = u,w ∈ E(G) : u ∈ U,w ∈W

eG(U,W ) size of EG(U,W ), eG(U,W ) = |EG(U,W )|NG(U,W ) set of all neighbours of U in W ,

NG(U,W ) = w ∈W : ∃u ∈ U s.t. u,w ∈ E(G)

degG(u,W ) degree of a vertex u in W , degG(u,W ) = |NG(u,W )|NG(U,W ) common neighbourhood of U in W ,

NG(U,W ) = w ∈W : u,w ∈ E(G) for all u ∈ U

2.2 Probabilistic tools and estimates

Lemma 2.1 (Markov’s Inequality). Let X be a non-negative randomvariable. For all t > 0 we have Pr[X ≥ t] ≤ E[X]

t .

We will use lower tail estimates for random variables which count thenumber of copies of certain graphs in a random graph. The followingversion of Janson’s inequality, tailored for graphs, will suffice. Thisstatement follows immediately from Theorems 8.1.1 and 8.1.2 in [AS04].

Theorem 2.2 (Janson’s inequality). Let p ∈ (0, 1) and consider a fam-ily Hii∈I of subgraphs of the complete graph on the vertex set [n]. LetG ∼ G(n, p). For each i ∈ I, let Xi denote the indicator random vari-


able for the event that Hi ⊆ G and, for each ordered pair (i, j) ∈ I × Iwith i 6= j, write Hi ∼ Hj if E(Hi) ∩ E(Hj) 6= ∅. Then, for

X =∑i∈I

Xi,

µ = E[X] =∑i∈I

pe(Hi),

δ =∑

(i,j)∈I×IHi∼Hj

E[XiXj ] =∑

(i,j)∈I×IHi∼Hj

pe(Hi)+e(Hj)−e(Hi∩Hj)

and any 0 < γ < 1,

Pr[X < (1− γ)µ] ≤ e−γ2µ2

2(µ+δ) .

Next, we state a couple of results concerning bounds on the number ofedges and neighbourhoods in random graphs. We use the following wellknown concentration bound.

Theorem 2.3 (Chernoff’s inequality). Let ε be a positive constant andX ∼ B(n, p) a binomial random variable with parameters n and p. Then

Pr[|X − EX| ≥ εEX] ≤ e−Ωε(EX).

The similar inequality for hypergeometric distributions was proven byChvátal [Chv79]. The following proposition is used in the proof of The-orem 6.3.

Proposition 2.4. For every µ > 0 there exists a constant C > 0,such that if p ≥ C/n then G ∼ G(n, p) a.a.s has the property thateG(X,Y ) = (1 ± µ)|X||Y |p for every disjoint X,Y ⊆ V (G) of size|X|, |Y | ≥ µn.

Proof. The proof follows from Theorem 2.3 (Chernoff’s inequality) andthe union bound over all possible X and Y . We omit the details.

The following proposition is used at the end of the proof of the Embed-ding Lemma (Lemma 6.17).

Proposition 2.5. For every d ∈ N and λ ∈ (0, 1) there exist constantC,L > 0, such that if

p ≥(C log n

n

)1/d

2.2. Probabilistic tools and estimates 19

then G ∼ G(n, p) a.a.s has the following property: for every familyD ⊆ 2V (G) of pairwise disjoint d-subsets we have∣∣∣∣∣ ⋃

S∈DN(S, V (G))

∣∣∣∣∣ ≥

(1± λ)|D|npd, if |D| ≤ λ/pd

(1± λ)n, if |D| ≥ L/pd.

Proof. Consider a (fixed) familyD ⊆ 2V (G) of pairwise disjoint d-subsetsof size |D| = x ≤ λ/pd and let U = V (G) \ ∪S∈DS. Observe thatthe bound on p implies |U | = n(1 − o(1)). For each v ∈ U , let Xv

be the indicator variable for the event that there exists S ∈ D withv ∈ N(S,U). Since all sets in D are pairwise disjoint, we have

Pr[Xv = 1] = 1− (1− pd)x ≥ xpd(1− λ/2) and Pr[Xv = 1] ≤ xpd.

We used the fact that (1 − a)b ≤ 1 − ab + (ab)2/2 for every b ∈ N anda constant a > 0 such that ab < 1 (follows from the binomial theorem).Therefore, for X =

∑v∈U Xv we have EX = nxpd(1±2λ/3). Note that

variables Xv are mutually independent, thus by applying Chernoff’sinequality we get

Pr

[∣∣ ⋃S∈D

N(S, V (G))∣∣ 6= (1± λ)nxpd

]≤ Pr[X 6= (1± 3λ/4)nxpd]

≤ e−αnxpd

≤ n−αCx,

where α > 0 depends only on λ. Taking a union bound over all possiblechoices for D, we obtain the desired upper-bound on the event thatthere exists a family D which violates the property of the lemma,

λ/pd∑x=1

nxdn−αCx = o(1),

for sufficiently large C = C(α, d).

To prove the second part, it suffices to only consider families of sizex = L/pd. Let D be a family of disjoint d-subsets with |D| = x andW ⊆ V (G) a subset of size |W | = bλnc. Then W ′ = W \

⋃S∈D S is of

size |W ′| = (1− o(1))|W |, thus

Pr[N(D,W ) = ∅] ≤ Pr[N(D,W ′) = ∅] = (1− pd)(1−o(1))λnx

≤ e−(1−o(1))λnxpd ≤ e−λnL/2,


with room to spare. Taking a union bound over all choices for a subsetW ⊆ V ′ of size |W | = bλnc and a family D, we obtain the follow-ing upper bound on the event that there exists D and W such thatN(D,W ) = ∅,(

n

bλnc

)nxde−λnL/2 ≤ 2nelogn·(L+1)/pd−λnL/2

≤ en+(L+1)n/C−λnL/2 = o(1),

for suitably chosen L = L(λ) > 0 and C ≥ L + 1. This implies|N(D, V ′)| ≥ n− bλnc for every family D of pairwise disjoint d-subsetsof size |D| ≥ L/pd, as required.

2.3 Graph density measures and decompo-sitions

Given an `-graph G for some integer ` ≥ 2, we use m`(G) to denote theso-called `-density defined as

m`(G) = maxJ⊆G,vJ>`

d`(J),

whered2(G) = (eG − 1)/(vG − `).

If m`(G) = d`(G) then we say that G is `-balanced, and if in additionm`(G) > d`(J) for every subgraph J ⊂ G with vJ ≥ `, we say thatG is strictly `-balanced. The notion of `-density plays a central role indetermining threshold for various properties considered in this thesis.

The following two statements make a connection between edge-densitymeasures ar(G) (the so-called arboricity of G) and m(G) (the densityof G) and certain decomposition properties.

Theorem 2.6 (Nash-Williams’ arboricity theorem [NW61, NW64]).Any graph G can be decomposed into dar(G)e edge-disjoint forests, where

ar(G) = maxG′⊆G

eG′

vG′ − 1.

The next lemma follows immediately from Hall’s theorem. For conve-nience of the reader we add its short proof.

2.3. Graph density measures and decompositions 21

Lemma 2.7. The edges of any graph G can be oriented such that themaximal outdegree is at most dm(G)e, where

m(G) = maxG′⊆G

eG′

vG′.

Proof. Let k := dm(G)e. We construct a bipartite graph G as follows.One vertex class consists of all edges of G (class Pe) and the other ofk copies of each vertex of G (class Pv). Furthermore, we add an edgebetween edge e and a vertex v if and only if v is an endpoint of e in G.It follows immediately from the definition of m(G) and the constructionthat G satisfies Hall’s condition with respect to the class Pe. Thus, Gcontains a matchingM that covers the set Pe. Orient an edge e = v, uof G towards u if e, v belongs to M (for some copy of v in Pv). Sinceeach vertex appears only k times in Pv, we deduce from the constructionthat the out-degree of each vertex is bounded by k. Since M covers Pe,this process describes the orientation of every edge.


Chapter 3Two selected proofs

3.1 A short proof of the Rödl-Ruciński The-orem – 1-statement

The following classic theorem of Rödl and Ruciński draws a connectionbetween Ramsey’s theorem and random graphs.

Theorem 3.1 (Rödl, Ruciński [RR93, RR94, RR95]). Let r ≥ 2 andlet H be a fixed graph that is not a forest of stars or, in the case r = 2,paths of length 3. Then there exist positive constants c = c(H, r) andC = C(H, r) such that

limn→∞

Pr[G(n, p)→ (H)r] =

0 if p ≤ cn−1/m2(H)

1 if p ≥ Cn−1/m2(H).

23

24 Chapter 3. Two selected proofs

In this section we give a short proof of the 1-statement, based on [NS16].The corresponding 0-statement is proven in Chapter 4.

For the exceptional case of a star with k edges it is easily seen that thethreshold is determined by the appearance of a star with r(k − 1) + 1edges. For the path P3 of length three the 0-statement only holds forp n−1/m2(P3) = n−1 since, for example, a C5 with a pendant edgeat every vertex has density one and cannot be 2-coloured without amonochromatic P3.

Observe that the expected number of copies of H in G(n, p) has theorder of nv(H)pe(H). On the other hand, the expected number of edgesof G(n, p) is in the order of n2p. Therefore, if nv(H)pe(H) ≤ cn2p forsome small constant c > 0, then we expect the copies of H to be looselyscattered – and finding a colouring that avoids the desired copy of Hshould be an easy task. Similarly, if nv(H)pe(H) ≥ Cn2p for some largeconstant C > 0, we expect that the copies of H overlap so heavilythat any colouring should contain the desired copy H. Actually, thesame argument holds for any subgraph of H and this thus motivatesthe definition of the 2-density (and more generally, the `-density) givenin Section 2.3.

The proof of the 1-statement requires two tools. The first one is a well-known quantitative strengthening of Ramsey’s theorem. We include itsshort proof for convenience of the reader.

Theorem 3.2 (Folklore). For every graph H and every constant r ≥ 2there exist constants α > 0 and n0 such that for all n ≥ n0, everyr-colouring of the edges of Kn contains at least αnvH monochromaticcopies of H.

Proof. From Ramsey’s theorem we know that there exists N := N(H, r)such that every r-colouring of the edges of KN contains a monochro-matic copy of H. Thus, in any r-colouring of Kn every N -subset of thevertices contains at least one monochromatic copy of H. As every copyof H is contained in at most

(n−vHN−vH

)N -subsets, the theorem follows e.g.

with α = 1/NvH .

We will need in particular the following easy consequence of Theo-rem 3.2.

Corollary 3.3. For every graph H and every r ∈ N there exist constantsn0 and δ, ε > 0 such that the following is true for all n ≥ n0. For any

3.1. A short proof of the Rödl-Ruciński Theorem – 1-statement 25

E1, . . . , Er ⊆ E(Kn) such that for all 1 ≤ i ≤ r the set Ei contains atmost εnv(H) copies of H, we have

|E(Kn) \ (E1 ∪ . . . ∪ Er)| ≥ δn2.

Proof. Let α and n0 be as given by Theorem 3.2 for H and r+1, and setε = α/2r. Further, let Er+1 := E(Kn)\ (E1∪ . . .∪Er) and consider thecolouring ϕ : E(Kn)→ [r+1] given by ϕ(e) = mini ∈ [r+1] : e ∈ Ei.By Theorem 3.2 there exist at least αnvH monochromatic copies ofH, ofwhich, by assumption on the sets Ei, at least 1

2αnvH must be contained

in Er+1. As every edge is contained in at most 2eHnvH−2 copies of H

the claim of the corollary follows e.g. for δ = α4eH

.

The second tool that we need is a consequence of the beautiful con-tainer theorems of Balogh, Morris and Samotij [BMS15] and Saxtonand Thomason [ST15]. The following theorem is from [ST15].

Definition 3.4. For a given set S and constants k, s ∈ N, let Tk,s(S)be the family of k-tuples of subsets defined as follows,

Tk,s(S) = (S1, . . . , Sk) | Si ⊆ S for 1 ≤ i ≤ k and |k⋃i=1

Si| ≤ s.

Theorem 3.5 (Theorem 2.3, [ST15]). For any graph H and ε > 0,there exist n0 and k > 0 such that the following is true. For everyn ≥ n0 there exist t = t(n), pairwise distinct k-tuples T1, . . . , Tt ∈Tk,kn2−1/m2(H)(E(Kn)) and sets C1, . . . , Ct ⊆ E(Kn), such that

(a) each Ci contains at most εnvH copies of H,

(b) for every H-free graph G on n vertices there exists 1 ≤ i ≤ tsuch that Ti ⊆ E(G) ⊆ Ci. (Here Ti ⊆ E(G) means that all setscontained in Ti are subsets of E(G).)

With these two tools in hand the proof of the 1-statement of Theorem 3.1is now easily completed.

Proof of Theorem 3.1 (1-statement). Let ε and δ be as in Corollary 3.3,and let n0 and k be as in Theorem 3.5 for H and ε, and assume that n ≥n0. If G(n, p) 9 (H)r, then there exists a colouring ϕ : E(G(n, p))→ rsuch that for all 1 ≤ j ≤ r the set Ej := ϕ−1(j) does not contain


a copy of H. By Theorem 3.5 we have that for every such Ej thereexists 1 ≤ ij ≤ t(n) such that Tij ⊆ Ej ⊆ Cij and Cij contains atmost εnvH copies of H. The trivial, but nonetheless crucial observationis that G(n, p) completely avoids E(Kn) \ (Ci1 ∪ . . . ∪ Cir ), which byCorollary 3.3 has size at least δn2.

Therefore we can bound Pr[G(n, p) 9 (H)r] by bounding the probabil-ity that there exist tuples Ti1 , . . . , Tir that are contained in G(n, p) suchthat E0(Ti1 , . . . , Tir ) := E(Kn) \ (Ci1 ∪ . . . ∪ Cir ) is edge-disjoint fromG(n, p). Thus

Pr[G(n, p) 9 (H)r] ≤∑

i1,...,ir

Pr[Ti1 , . . . , Tir ⊆ G(n, p) ∧

G(n, p) ∩ E0(Ti1 , . . . , Tir ) = ∅],

where i1, . . . , ir run over the choices given by Theorem 3.5. Note thatthe two events in the above probability are independent and the prob-ability can thus be bounded by

p|⋃rj=1 T

+ij| · (1− p)δn

2

,

where by T+ij

we denote the union of the sets of the k-tuple Tij . Thesum can be bounded by first deciding on

s := |r⋃j=1

T+ij| ≤ r · kn2−1/m2(H),

then choosing that many edges (((n2)s

)choices) and finally deciding for

every edge in which sets of the k-tuples Tij it appears ((2rk)s choices).Together, this gives

Pr[G(n, p) 9 (H)r] ≤ (1− p)δn2

·rkn2−1/m2(H)∑

s=0

((n2

)s

)(2rk)sps

≤ e−δn2p ·

rkn2−1/m2(H)∑s=0

(e2rkn2p

2s

)s.

Since the Ramsey property is monotone increasing we may assume p =Cn−1/m2(H). By choosing C sufficiently large (with respect to k) wehave

rkn2−1/m2(H)∑s=0

(e2rkn2p

2s

)s≤ n2 ·

(e2rkC

2rk

)(rk/C)n2p

≤ e 12 δn

2p

3.2. Universality for degenerate graphs 27

and thus Pr[G(n, p) 9 (H)r] = o(1), as desired.

The same approach, with Theorem 3.2 and Theorem 3.5 replaced withthe corresponding hypergraph versions gives an alternative proof for thecorresponding result of Friedgut, Rödl and Schacht [FRS10] and Conlonand Gowers [CG10].

3.2 Universality for degenerate graphs

In this section we demonstrate one of the main ideas used in the proofof the Embedding Lemma (Lemma 6.17) in Chapter 6. For integers∆, D, n ∈ N, let D∆(n,D) denote the family of all D-degenerate graphson at most n vertices and with maximum degree at most ∆. We areinterested in determining the threshold for the property that G(n, p) isD∆((1 − ε)n,D)-universal. This is usually referred to as the almost-spanning universality.

The case D = 1 was resolved by Alon, Sudakov and Krivelevich [AKS07]and indepndently Balogh, Csaba, Pei and Samotij [BCPS10]. UsingTheorem 4.1 in [FNP16] it is easy to show that G(n, p) is almost surelyD∆((1 − ε)n,D)-universal if p ≥ n−1/(2D)+o(1). On the other hand, asimple first-moment argument gives the lower bound of order n−1/D.This can be seen, for example, by considering the D-th power of a longpath. We show that this value can be achieved up to the logarithmicfactor.

Theorem 3.6. For any constant ε > 0 and (fixed) integers ∆, D ∈ Nwith D ≤ ∆, there exists a constant C > 0 such that if

p ≥(C log2 n

n log log n

)1/D

then G(n, p) is a.a.s D∆((1− ε)n,D)-universal.

Proof. Let L and C be constants larger than ones given by Proposition2.5 with λ ← ε/2 and d taking every value in 1, . . . , D. By a unionbound we obtain that G ∼ G(n, p) a.a.s satisfies the property given byProposition 2.5 for all d ∈ 1, . . . , D. Let

k =

⌈2 log n

log log n

⌉and s = dεn/(3k)e,


and consider a random partition V 1, . . . , V k of V (G) with |Vi| = s fori ∈ 2, . . . , k and |V1| = n − sk ≥ (1 − ε/2)n. Given a family Dof pairwise disjoint d-subsets of V (G) of size at most λ/pd and j ∈2, . . . , k, the expected size of ∪S∈DNG(S, V j) is

E

[∣∣ ⋃S∈D

NG(S, V j)∣∣] =

∣∣∣∣∣ ⋃S∈D

NG(S)

∣∣∣∣∣ · sn≥ (1− λ)|D|npd · ε

3k≥ C ′|D| log n,

where C ′ = C ′(C, ε, λ) grows with C. Therefore, Chernoff’s inequalityfor hypergeometric distributions and a union bound over all families Dand indices j ∈ 2, . . . , k show that there exists a partition V 1, . . . , V k

with the following property: for every d ∈ [D], j ∈ [k] and every familyD of disjoint d-subsets it holds that∣∣∣∣∣ ⋃

S∈DNG(S, V j)

∣∣∣∣∣ >ε|D|npd

6k , if |D| ≤ λ/pd and j ∈ 2, . . . , k(1− ε)n, if |D| ≥ L/pd and j = 1.

(3.1)In the first inequality we have implicitly assumed λ < 1/2 and thesecond one follows from |V1| ≥ (1− ε/2)n.

We now describe how this implies D∆(`,D)-universality of G, where` = (1 − ε)n. Consider an arbitrary graph H ∈ D∆(`,D) and leth1, h2, . . . , h` be an ordering of the vertices of H such that

degH(hi, h1, . . . , hi−1) ≤ D (3.2)

for every i ∈ [`]. We iteratively find distinct vertices v1, . . . , v` ∈ V (G)such that the natural mapping f : V (H)→ V (G) given by f(hi) = vi isan embedding. For each i = 1, . . . , `, choose vi as follows:

(i) if Si := NH(hi, h1, . . . , hi−1) is empty set then choose an arbi-trary vi ∈ V 1 \ v1, . . . , vi−1;

(ii) otherwise, let ji ∈ 1, . . . , k be the smallest number such that

Ci := NG(f(Si), Vji) \ v1, . . . , vi−1

is non-empty and choose arbitrary vi ∈ Ci.

Assuming that the procedure is well-defined (that is, it is indeed possibleto make choices as stated), definitions of Si and Ci imply that the

3.2. Universality for degenerate graphs 29

resulting sequence of vertices induces an embedding of H into G. Wenow verify that steps (i) and (ii) are always possible.

From |V 1| ≥ (1− ε/2)n > ` we have that (i) is well-defined. In order toprove that the step (ii) is possible, we show

(∗) |V j ∩ v1, . . . , v`| ≤ g(j) =

(1− ε)n, if j = 1

LD2∆/pD, if j = 2g(j−1)D2∆6k

εnpD, otherwise,

for every j ∈ 1, . . . , k. Then owing to the choice of k we haveg(k) → 0, which together with (3.1) applied on D = f(Si) impliesNG(f(Si), V

k) \ v1, . . . , vi−1 6= ∅ for every i ∈ [`]. Consequently ji in(ii) is well-defined.

Since v(H) = (1 − ε)n, (∗) trivially holds for j = 1. Let us assume,towards the contradiction, that there exists j ∈ 2, . . . , k such that theset of indices J = i ∈ [`] : vi ∈ V j is of size |J | > g(j) and, moreover,consider the smallest such j. From |Si| ≤ D (follows from (3.2)) andthe fact that H has a maximum degree at most ∆, it is easy to see (forexample, using a greedy procedure) that there exists d ∈ 1, . . . , D anda subset J ′ ⊆ J of size

|J ′| ≥ |J |D2∆

>g(t)

D2∆≥ st :=

L/pd, if t = 2g(j−1)6kεnpd

, if t ≥ 3,

such that |Si| = d and Si ∩ Si′ = ∅ for distinct i, i′ ∈ J ′. Moreover, wecan assume that J ′ is exactly of size |J ′| = sj and observe that if j ≥ 3then sj = o(1/pd). Therefore, from (3.1) applied with D = Sii∈J′ weinfer ∣∣∣∣∣ ⋃

i∈J′NG(Si, V

j−1)

∣∣∣∣∣ > g(j − 1).

On the other hand, by the definition of ji in (ii) we have

NG(f(Si), Vj−1) ⊆ v1, . . . , vi−1

for every i ∈ J ′, which implies⋃i∈J′

NG(Si, Vj−1) ⊆ V j−1 ∩ v1, . . . , v`.

However, by the minimality of j we have |V j−1∩v1, . . . , v`| ≤ g(j−1)which gives a contradiction. This finishes the proof of the lemma.


Chapter 4Ramsey-type problems in random

graphs

In this chapter we introduce a general framework for proving lowerbounds for various Ramsey type problems within random settings. Themain idea is to view the problem from an algorithmic perspective: weaim at providing an algorithm that finds the desired colouring with highprobability. Our framework allows to reduce the probabilistic problem ofwhether the Ramsey property at hand holds for random (hyper)graphswith edge probability p to a deterministic question of whether thereexists a finite graph that forms an obstruction.

Proofs presented in this chapter are based on [NS16, NŠS15a, NSS15b]and yet unpublished [NPŠS14].

31

32 Chapter 4. Ramsey-type problems in random graphs

4.1 Results – old and new

Ramsey’s theorem for random `-graphs

The systematic study of Ramsey properties of random graphs was ini-tiated by Łuczak, Ruciński and Voigt [ŁRV92] in the early nineties.Shortly thereafter Rödl and Ruciński determined the threshold func-tion of the graph Ramsey property for all graphs H (Theorem 3.1). Itis worth noting that in the case when H is a triangle, Friedgut, Rödl,Ruciński and Tetali [FRRT06] have strengthened Theorem 3.1 by show-ing that there exists a sharp threshold.

Extending Theorem 3.1 to hypergraphs, Rödl and Ruciński [RR98]proved that for the 3-uniform clique on 4 vertices and 2 colours the 1-statement is determined by the 3-density as one would expect, followingthe intuition given in Section 3.1. They also conjectured that, similarlyto the graph case, the threshold should be determined by the `-densityfor “most” of the `-graphs H. Rödl, Ruciński and Schacht [RRS07] latershowed that the 1-statement holds for all `-partite `-graphs. In fullgenerality the 1-statement was resolved only recently by Friedgut, Rödland Schacht [FRS10] and independently by Conlon and Gowers [CG10].

Theorem 4.1 ([FRS10, CG10]). Let H be an `-graph with maximumdegree at least 2 and let r ≥ 2. Then there exists a constant C > 0 suchthat for p ≥ Cn−1/m`(H) we have

limn→∞

P[G(`)(n, p)→ (H)r] = 1.

As already mentioned, the proof presented in Section 3.1 extends to thehypergraph case in a straightforward way, thus giving an alternativeproof of Theorem 4.1.

The matching lower bound for Theorem 4.1 in case where H is a hyper-graph clique (and some other special cases) was obtained in [Tho13].

Theorem 4.2. Let k, ` ∈ N be such that 2 ≤ ` < k and let r ≥ 2. Thenthere exist constants c, C > 0 such that

limn→∞

Pr[G(`)(n, p)→ (K(`)k )r] =

1, if p ≥ Cn−1/m`(K

(`)k ),

0, if p ≤ cn−1/m`(K(`)k ).

We will deduce 0-statement of Theorem 3.1 (except for a few special

4.1. Results – old and new 33

cases) and Theorem 4.2 as a straightforward corollary of Theorem 4.4,presented in the next section.

Anti-Ramsey property for r-bounded colourings

If we allow colourings with an unbounded number of colours we arriveat the so-called anti-Ramsey problem where we are interested in findinga rainbow copy of H, i.e., a copy of H in which each edge uses a differentcolour. Again, to avoid trivialities one needs to forbid colourings withtoo few colours. This has been done in several different ways. Here weinsist that each colour is used at most r times (we call this an r-boundedcolouring). We write

Ga-ram−−−→r

H

if every r-bounded edge colouring of G contains a rainbow copy of H.

Lefmann, Rödl andWysocka [LRW96] considered the following question.Given a complete graph Kn with edges colored using an r-boundedcolouring, what is the largest k such that G contains a rainbow copyof Kk. Bohman, Frieze, Pikhurko and Smyth [BFPS10] initiated thestudy of a similar question in G(n, p). The authors proved that givena graph H and a constant r ≥ r(H), the threshold for the property ofbeing r-bounded anti-Ramsey matches the intuition.

Theorem 4.3 ([BFPS10]). Let H be a graph which contains a cycle.Then there exists a constant r0 = r0(H) such that for each r ≥ r0(H)there exist constants c, C > 0 and

limn→∞

Pr[G(n, p)a-ram−−−→r

H] =

1, if p ≥ Cn−1/m2(H),

0, if p ≤ cn−1/m2(H).

It is easy to see that for the case H = K3 and 2-bounded colouringsthere exists an obstruction, namely the complete graph on 4 vertices. Werefer the reader to [BFPS10] for details regarding the results in the caseH = K3. For other graphs H it is not obvious whether the restrictionon r is really needed. Indeed, the following theorem strengthens the0-statement of Theorem 4.3 by showing that r = 2 actually suffices formost cases. In part (iii) we also provide an extension to hypergraphs inthe case of cliques.

Theorem 4.4. Let ` ≥ 2 and H be an `-graph. Let H ′ ⊆ H be a strictly`-balanced subgraph such that m`(H

′) = m`(H) . Then there exists a


constant c > 0 such that G ∼ G(`)(n, p) w.h.p. satisfies Ga-ram−−−−→

2H if

one of the following holds,

(i) ` = 2, H ′ contains a cycle, H ′ K3, C4 and p ≤ cn−1/m2(H),or

(ii) ` = 2, H ′ ∼= C4 and p n−1/m2(C4), or

(iii) ` ≥ 3, r ≥ `+ 1 and (`, r) 6= (3, 4),H ′ ∼= K

(`)r and p ≤ cn−1/m`(K

(`)r ).

Anti-Ramsey property for proper edge colourings

We writeG

a-ram−−−→prp

H

if every proper edge colouring of G contains a rainbow copy of H.

The first result on the relation between random graphs and the proper-colouring version of the anti-Ramsey property comes from the followingquestion raised by Spencer: is it true that for every g there exists a graphG with girth at least g such that G a-ram−−−→

prpC` for some `. The question

was answered in positive by Rödl and Tuza [RT92]. They proved thatfor every ` there exists some sufficiently small p = p(n) such that w.h.p.G(n, p)

a-ram−−−→prp

C`. Only much later, Kohayakawa, Kostadinidis andMota [KKM11, KKM14a] started a systematic study of this property inthe random settings. In particular, they proved that the upper boundis as expected.

Theorem 4.5 ([KKM14a]). Let H be a graph. Then there exists aconstant C > 0 such that for p ≥ Cn−1/m2(H) we have

limn→∞

Pr[G(n, p)a-ram−−−→prp

H] = 1.

Note that H = K3 is a trivial case since K3 is an obvious obstruction.Therefore, any graph H which contains K3 as the 2-densest subgraph isa potential candidate for having an obstruction. Indeed, the above au-thors showed in [KKM14b] that there exists an infinite family of graphsfor which the threshold is asymptotically below the guessed one. Herewe prove that at least in the case of sufficiently large complete graphsand cycles, the situation is as expected.

4.1. Results – old and new 35

Theorem 4.6. Let H be a graph isomorphic to either a cycle on atleast 7 vertices or a complete graph on at least 19 vertices. Then thereexist constants c, C > 0 such that

limn→∞

Pr[G(n, p)a-ram−−−→prp

H] =

1, if p ≥ Cn−1/m2(H),

0, if p ≤ cn−1/m2(H).

We remark that our bounds on the minimum size of the cliques resp.cycles are simply a consequence of our proof and probably not tight. Asfar as we know, the result actually could hold for all cliques and cyclesof size at least 4.

Maker-Breaker H-game

We briefly repeat the definition of the Maker-Breaker H-game given inSection 1.2.2. Given a (large) graph G and a (small) graph H, the H-game on G is played on the edges of G and the winning sets are theedge sets of all copies of H appearing in G as subgraphs. Maker andBreaker alternately claim unclaimed edges of the graph G until all theedges are claimed. Maker wins if he claims all the edges of a copy of Hin which case we say that the graph G is Maker’s win, and otherwiseBreaker wins, in which case we say that the graph G is Breaker’s win.

Here we are interested in the threshold for the property that G ∼G(n, p) is Maker’s win. Improving upon results of Stojaković and Szabó[SS05] and Müller and Stojaković [MS14], the following was obtained in[NSS15b] using an argument similar to those in Section 4.3.

Theorem 4.7. Let H be a graph for which there exists H ′ ⊆ H suchthat d2(H ′) = m2(H), H ′ is strictly 2-balanced and it is not a tree or atriangle. Then there exist constants c, C > 0 such that

limn→∞

Pr[G(n, p) is Maker’s win in the H-game] =

=

1, p ≥ Cn−1/m2(H),

0, p ≤ cn−1/m2(H).

Rather than using a proof from [NSS15b], we show that the Makerproperty is “sandwiched” between Ramsey and anti-Ramsey properties.

First, note that the property of not being 2-bounded anti-Ramsey for His stronger than not having the Maker property. Indeed, assume that a


graph G is such that there exists a 2-bounded colouring of G without arainbow copy of H. Then Breaker can apply the following strategy: fixone such 2-bounded colouring and whenever Maker claims an edge, claimthe other edge with the same colour. Maker’s graph then corresponds toa rainbow subgraph of G and thus does not contain a copy of H. Thisshows that the threshold for the Maker property is at least as largeas the corresponding 2-bounded anti-Ramsey property. On the otherhand, if G is Ramsey for H then the strategy stealing argument impliesthat Maker has a winning strategy. Thus the threshold for the Makerproperty is at most the threshold for the Ramsey property.

In particular, if H is such that the threshold for anti-Ramsey and Ram-sey property are asymptotically the same, then we also know the thresh-old for the Maker property in the H-game. Therefore, Theorem 4.4 fullyimplies Theorem 4.7, with the exception in the case when the 2-densestsubgraph of H is C4. In this case Theorem 4.4 requires that p is asymp-totically below n−1/m2(C4) whereas Theorem 4.7 allows p to be of thesame order. For discussion of cases of H not covered by Theorem 4.7,refer to [NSS15b].

4.1.1 Outline of the proof

The main goal of this chapter is to provide a unifying framework forproving 0-statements for Ramsey-type properties. The main idea is toview the problem from an algorithmic perspective: we aim at providingan algorithm that finds the desired colouring with high probability. Todo this we take the given random hypergraph G(`)(n, p) as input andfirst ‘strip of’ easily colourable edges, where the definition of ‘easilycolourable’ depends on the type of the given Ramsey problem. We thenargue that whatever remains after the end of this stripping procedurecan be partitioned into blocks that can be coloured separately. Ourkey result (Theorem 4.12) states that with probability 1 − o(1) theseblocks will have size at most some constant L that depends (in somewell-understood way) on the graph H. It is well known that in a typicalrandom hypergraph with density n−α all subgraphs of constant sizehave density at most 1/α. This implies that it suffices to prove that astatement of the form

all `-graphs G with m(G) ≤ m`(H) satisfy G∗−−→ H (4.1)

4.2. A general framework 37

holds deterministically, where by ∗−−→ we mean any of the discussedRamsey properties. Note that any graph with density m`(H) appearsin G(`)(n, p) with constant probability for p = cn−m`(H) (cf. proofof Corollary 4.13 for details). Thus, the condition in (4.1) is actuallynecessary for the 0-statement to hold. Formally, we call a graph G anobstruction for H if m(G) ≤ m`(H) and G

∗−−→ H. Note that suchobstructing graphs G indeed do exist. For some Ramsey type problemsthere are only a few, for others there exist infinitely many. We commenton that in more detail later. Our aim is to show that the conditionin (4.1) is also sufficient, i.e. in order to prove the 0-statement it issufficient to show that obstructions do not exist. We summarize this inthe following “meta-theorem”.

Meta-Theorem. Let H be an `-graph for which (4.1) holds. Then

limn→∞

Pr[G(`)(n, p)∗−−→ H] =

1, if p ≥ Cn−1/m`(H),

0, if p ≤ cn−1/m`(H).

Recall from the previous section that the 1-statements are known to holdfor all Ramsey problems considered in this chapter. The key statementof our meta theorem is thus that the bound from the 1-statement isactually tight, whenever (4.1) holds.

In Section 4.2 we prove our framework theorem, Theorem 4.12. InSection 4.3 we provide the proofs for Theorems 4.2, 4.4 and 4.6 byshowing deterministic statements corresponding to (4.1).

4.2 A general framework

4.2.1 Outline of the method

The key idea for the proof of the Meta-Theorem from Section 4.1.1 is tointroduce appropriate notions that capture the structure of overlappingcopies of H. In the following definitions we always assume that Hcontains at least two edges.

Definition 4.8 (H-equivalence). Given `-graphs H and G, we say thattwo edges e1, e2 ∈ E(G) are H-equivalent, with notation e1 ≡H e2, if forevery H-copy H ′ in G we have e1 ∈ E(H ′) if and only if e2 ∈ E(H ′).


Definition 4.9. Given an `-graph H we define γ(H) to be the largestintersection of two distinct edges in H, i.e.

γ(H) := max|e1 ∩ e2| : e1, e2 ∈ E(H) and e1 6= e2.

Definition 4.10 (H-closed property). For given `-graphs H and G, wedefine the property of being H-closed as follows:

• an edge e ∈ E(G) is H-closed if

(a) γ(H) = `− 1 and e belongs to at least two H-copies in G or(b) γ(H) < ` − 1 and e belongs to at least two H-copies in G

and no edge e′ ∈ E(G) \ e is H-equivalent to e,

• an H-copy H ′ in G is H-closed if at least three edges from E(H ′)are closed,

• the `-graph G is H-closed if every vertex and edge of G belongsto at least one H-copy and every H-copy in G is closed.

If the `-graph H is clear from the context, we simply write closed.

Definition 4.11 (H-blocks). Given `-graphs H and G such that G isH-closed, we say that G is an H-block if for every non-empty propersubset of edges E′ ( E(G) there exists an H-copy H ′ in G such thatE(H ′) ∩ E′ 6= ∅ and E(H ′) \ E′ 6= ∅ (in other words, there exists anH-copy which partially lies in E′).

With these definitions at hand we can now formulate our key result:

Theorem 4.12 (Framework theorem). Let ` ≥ 2 be an integer and H astrictly `-balanced `-graph such that either H has exactly three edges andγ(H) = `− 1 or H contains at least 4 edges. Then there exist constantsc, L > 0 such that for p ≤ cn−1/m`(H), G ∼ G(`)(n, p) satisfies w.h.p.that every H-block B ⊆ G contains at most L vertices.

In all our applications we will use the following corollary of Theorem4.12 which gives a bound on the density m of H-blocks.

Corollary 4.13. Let ` ≥ 2 be an integer and H a strictly `-balanced`-graph such that either γ(H) = ` − 1 and H contains at least 3 edgesor H contains at least 4 edges. Then there exists a constant c > 0 suchthat for p ≤ cn−1/m`(H), G ∼ G(`)(n, p) w.h.p. satisfies that for everyH-block B ⊆ G we have m(B) ≤ m`(H). Moreover, if p n−1/m`(H)

then strict inequality holds.


We conclude this section with a basic property of H-closed graphs thatwill be used throughout the applications.

Lemma 4.14. Let H be an `-graph. Then if an `-graph G is H-closed,there exists a partitioning E(G) = E1 ∪ . . . ∪ Ek, for some k ∈ N, suchthat each subgraph Bi induced by the set of edges Ei is an H-block andeach H-copy in G is entirely contained in some block Bi.

Proof. Let G be an H-closed `-graph and consider a smallest non-emptysubset of edges E′ ⊆ E(G) such that every H-copy is either completelycontained in E′ or avoids edges in E′. Observe that if an H-copy H ′ inG contains an edge e ∈ E′, then by the choice of E′ we have E(H ′) ⊆ E′.Similarly, if an H-copy H ′ in G contains an edge e ∈ E(G) \ E′ thenE(H ′) ⊆ E(G) \ E′. Therefore, every edge e ∈ E′, resp. e ∈ E(G) \ E′which was H-closed in G remains H-closed in G[E′], resp. G \E′, thusboth G[E′] and G \E′ are H-closed. By the minimality of E′ it followsthat G[E′] is an H-block. We can now set E1 := E′ and repeat theprocedure on G′ := G \ E′. In this way we obtain the desired partitionE1, . . . , Ek.

4.2.2 Proof of the framework theorem

Here we show that H-blocks are with high probability only of constantsize (Theorem 4.12). Before we prove Theorem 4.12, we first show howit implies Corollary 4.13.

Proof of Corollary 4.13. Let L and c be constants given by Theorem4.12 when applied to an `-graph H. Without loss of generality, we mayassume that c < 1. We first consider the case p ≤ cn−1/m`(H).

Let α ∈ R be a strictly positive constant such that for every `-graph Son at most L vertices with m(S) > m`(H) we have m(S) ≥ m`(H) +α.More formally, we define an α > 0 as follows,

α := minm(S)−m`(H) | v(S) ≤ L and m(S) > m`(H).

Since there are only finitely many such `-graphs S, α is well-defined.Consider now some `-graph S on at most L vertices with m(S) ≥m`(H)+α and let S′ ⊆ S be a subgraph such that e(S′)/v(S′) = m(S).Let XS′ be the random variable which denotes the number of S′-copies


in G. Then the expected number EXS′ of S′-copies in G ∼ G(`)(n, p) isat most

EXS′ ≤ nv(S′)pe(S′) ≤ nv(S′)−e(S′)/m`(H)

=(n1−m(S′)/m`(H)

)v(S′)

≤ n−α·v(S′)/m`(H) = o(1).

Therefore, by Markov’s inequality (Lemma 2.1) we have

Pr[G contains an S-copy] ≤ Pr[G contains an S′-copy]

= Pr[XS′ ≥ 1] ≤ EXS′ .

As there exist less than 2(L`) different `-graphs on at most L vertices, aunion-bound over all such `-graphs thus also gives

Pr[∃S ⊆ G such that v(S) ≤ L and m(S) > m`(H)] = o(1).

In particular, since w.h.p. G is such that every H-block B ⊆ G containsat most L vertices it follows that m(B) ≤ m`(H), as required.

Let us now assume that p n−1/m`(H). Similarly as in the previouscase, if S is an `-graph on at most L vertices with m(S) ≥ m`(H), thenfor p n−1/m`(H) we have that the expected number of S′-copies is

EXS′ ≤ nv(S′)pe(S′) = o(nv(S′)−e(S′)/m`(H)) = o(1),

where S′ ⊆ S is such that e(S′)/v(S′) = m(S). The same argument asbefore shows that G contains no copy of S, which finishes the proof.

Proof of Theorem 4.12. Our proof is a generalization of the approachfrom [NS16] to hypergraphs and general Ramsey problems. The proofis essentially a first moment argument. We enumerate all possible H-blocks on more than L vertices and show that the probability that oneor more of them appears in G ∼ G(`)(n, p) is o(1). The difficulty liesin the fact that straightforward enumerations (like choosing subsets ofedges) do not work: we have too many choices. We thus have to designa more efficient way to encode H-blocks. To do that we make use ofAlgorithm 1 that enumerates H-copies of a block in some clever way.

Let B be an H-block. Algorithm 1 maps B to a sequence (H0, . . . ,Hs)of copies of H. In order to see that the algorithm is well-defined itsuffices to show that lines 9 and 11 can always be executed. For line


1 H0 ← an arbitrary H-copy in B2 G0 ← H0

3 i← 04 while Gi 6= B do5 i← i+ 16 if Gi−1 contains an H-copy which is not closed then7 j ← smallest index j < i such that Hj is not closed8 e← an edge in Hj which is not closed in Gi−1 but closed in B9 Hi ← an H-copy in B but not Gi−1 which contains e

10 else11 Hi ← an arbitrary H-copy in B but not Gi−1 which intersects

Gi−1 in at least one edge12 end13 Gi ← Gi−1 ∪Hi

14 end15 s← i

Algorithm 1: Construction of a grow sequence for an H-block B.

9 this follows directly from the condition in the if-statement: an H-copy that is not yet closed contains an edge e that is closed in B butnot yet in Gi−1. As in line 8 we choose exactly such an edge, thedesired copy in line 9 exists. Similarly, if at some point the executionof line 11 would not be possible, this would imply that there exists asubgraph Gi ( B such that every H-copy in B is completely containedin either Gi or Gi = B \E(Gi). Since Gi is non-empty (it contains H0)this contradicts the assumption that B is an H-block. Thus line 11 iswell-defined. Finally, as the number of edges in Gi increases with eachiteration and E(Gi) ⊆ E(B), at some point Gi will be equal to B andthe algorithm will stop.

Note that the sequence (H0, . . . ,Hs) fully describes a run of the algo-rithm. We call it a grow sequence for B and each Hi in it a step of thesequence, 0 ≤ i ≤ s. Given some grow sequence S := (H0, . . . ,Hs) forB we can easily reconstruct B as the union of all Hi, 0 ≤ i ≤ s. We nowturn to the question of how to enumerate such sequences efficiently.

Let us fix an arbitrary labelling of the vertices of H, say V (H) =w1, . . . , wv(H). Every H-copy in B can be specified by an injectivemapping f : V (H) → V (B), thus we can represent every H-copy in Bas a v(H)-tuple of vertices of B where the i-th element of the tuple de-


termines f(wi), for 1 ≤ i ≤ v(H). Accordingly, we could represent everygrow sequence as a sequences of v(H)-tuples of vertices in V (B). Unfor-tunately, such an encoding is still too inefficient. We improve on this byusing the fact that every H-copy Hi from a grow sequence (H0, . . . ,Hs)has a non-empty intersection with H0 ∪ . . . ∪Hi−1 . We now make thismore precise.

We distinguish three step types. We call H0 the first step. For i ≥1 we call the step Hi regular if the intersecting subgraph Gi−1 ∩ Hi

corresponds to exactly one edge, and degenerate otherwise. In the firstmoment argument that we elaborate on below we choose the type of eachstep (regular or degenerate). For each type we then have to multiplythe number of choices by the probability that the new edges (the edgesin E(Hi) \ E(Gi−1)) are present in G.

For a regular step Hi created in line 9, the intersection with Gi−1 corre-sponds exactly to a non-closed edge e in Hj , where j < i is the smallestindex j < i such that Hj is not closed. Note that the index j can beuniquely reconstructed from the graph Gi. That is, we do not have tochoose it. This edge can be chosen in e(H) ways. Furthermore, wehave to choose which vertices in Hi correspond to these vertices, givinganother factor of v(H)`. It remains to choose the other v(H) − ` newvertices of Hi, which in turn describe the e(H)− 1 new edges that arerequired to be present. The total contribution of such a step is thus

e(H)v(H)`nv(H)−`pe(H)−1 ≤ e(H)v(H)`ce(H)−1 ≤ c < 1, (4.2)

where c is the constant in Theorem 4.12 which we choose small enoughfor the above to hold.

In contrast to regular steps created in line 9, if a regular step Hi iscreated in line 11 then a copy Hj which contains an intersecting edge ofHi and Gi−1 is not fully determined by Gi−1 and we need to choose it.By construction, the `-graph Gi−1 contains at most v(H)·i vertices, thusthere are at most (v(H) · i)` choices for the vertices in the attachmentedge in Gi−1 and the contribution of such a step is

e(H)(v(H) · i)`nv(H)−`pe(H)−1(4.2)≤ i`, (4.3)

again using the assumptions on the choice of c in (4.2)).

Now consider the case of degenerate steps, i.e. those for which H ′ :=Hi ∩ Gi−1 satisfies v(H) > `. We can choose which vertices of Gi−1


correspond to H ′ in (v(H) · i)v(H) many ways. Furthermore, since H isstrictly `-balanced and H ′ ( H we have

e(H ′)− 1

v(H ′)− `<e(H)− 1

v(H)− `= m`(H)

and thus

e(H)− e(H ′)v(H)− v(H ′)

=(e(H)− 1)− (e(H ′)− 1)

(v(H)− `)− (v(H ′)− `)> m`(H). (4.4)

This implies that we can choose a constant α > 0 such that for allH ′ ( H with v(H) > ` it holds that

v(H)− v(H ′)− e(H)− e(H ′)m`(H)

< −α.

Applying this to a degenerate step Hi, we obtain that the contributionis upper-bounded by∑

H′(Hv(H′)>`

(v(H ′) · i)v(H′)nv(H)−v(H′)pe(H)−e(H′)

≤ iv(H) · cn−α∑H′(Hv(H′)>`

v(H)v(H′)

≤ iv(H)n−α, (4.5)

where we again assume that c is chosen small enough for the above tohold.

Thus, degenerate steps introduce a factor iv(H)n−α, which suggests thatsequences containing many of them are very unlikely to appear in G.Similarly, regular steps created in line 9 introduce a factor of c < 1,which suggests that sequences containing Θ(log n) of these steps arealso unlikely to appear in G. The next claim provides bounds on thenumber of degenerate and regular steps created in line 11 that will allowus to conclude the proof.

Claim 4.15. Let S = (H0, . . . ,Hs) be a grow sequence correspondingto an execution of Algorithm 1. Then the following holds:

(a) If S contains at most d degenerate steps, then s ≤ 3d · v(H).


(b) If a prefix S′ of S contains at most d degenerate steps, then everyregular step Hj in S′, with j ≥ 3d · v(H) + 2, is created in line 9.

Intuitively, what Claim 4.15 tells us is that in a long grow sequenceeither there will be many degenerate steps or most of the steps willbe regular steps created in line 9. Note that every degenerate step, asEquation (4.5) shows, introduces a factor of Θ(n−α+o(1)) to the expec-tation of the number of appearances of S (for i = O(log n)) and regularstep created in line 9 introduces a constant factor c < 1. We defer theformal proof of Claim 4.15 to the next section.

With the help of Claim 4.15 we can now finish our first moment ar-gument. Set dmax := v(H)/α + 1 and L := 3dmaxv(H) + 1 and letS = (H0, . . . ,Hs) be a grow sequence of length more than L. ByClaim 4.15(a) every such sequence S must contain at least dmax de-generate steps. We now distinguish two cases. Let sd be the step inwhich the dmax-th degenerate step occurs in S. If sd < smax, wheresmax := v(H) log n + dmax + L, then we set S′ := (H0, . . . ,Hsd). Oth-erwise, we set S′ := (H0, . . . ,Hsmax

). We prove that in both cases theexpected number of possible grow sequences S longer than L which havea prefix S′ is o(1).

Observe that, in any case, S′ is a prefix of S that contains at mostdmax degenerate steps. Then, by Claim 4.15(b), if Hi is a regular stepfrom S′ created in line 11, we have i ≤ L. Let us first consider thecase when the dmax-th degenerate step occurs before step smax, that issd ∈ dmax, . . . , smax−1. For a fixed such sd there are

(sd−1

dmax−1

)ways to

choose steps in which the first dmax− 1 degenerate steps have occurred.We can now upper bound the expected number of such sequences S′ asfollows

smax−1∑sd=dmax

(sd−1dmax−1

)nv(H)

(sv(H)d n−α︸︷︷︸eq. (4.5)

)dmax((L)`︸︷︷︸

eq. (4.3)

)L= polylog(n) · nv(H)n−α·dmax = o(1).

Here we bound the contribution of the first step by nv(H), drop thecontribution of c < 1 for all regular steps created in line 9, and use thefact that only the first L + 1 steps can be regular steps created in line11.

Let us now consider the case sd ≥ smax. Note that then there ared ∈ 0, . . . , dmax degenerate steps within the first smax steps. Similarly


as in the previous case, we can upper bound the expected number ofsuch sequences S′ as follows:

dmax∑d=0

(smax

d

)nv(H)

(sv(H)

max n−α︸︷︷︸

eq. (4.5)

)d((L)`︸︷︷︸

eq. (4.3)

)Lcsmax−d−L︸︷︷︸

eq. (4.2)

= polylog(n) · nv(H) · csmax−d−L

= polylog(n) · 2v(H) logncv(H) logn = o(1),

where we used the fact that c is small enough and in particular smallerthan 1/2.

We can now conclude that the probability that G contains a possiblegrow sequence S of length longer than L as follows

Pr[S of length at least L] ≤Pr[S contains a prefix S′ as described] = o(1),

where the last inequality follows from Markov’s inequality. Thus, withprobability 1− o(1), every H-block in G contains at most v(H) · (L+ 1)vertices.

Proof of Claim 4.15

Let Si := (H0, . . . ,Hi), for 0 ≤ i ≤ s. For any Si and any regular stepHj , j ≤ i we call the edge e := E(Gj−1) ∩ E(Hj) the attachment edgeof Hj and the vertices in V (Hj)\V (Gj−1) the inner vertices of Hj . Forj ≤ i, we say that a regular step Hj is fully-open in Si if

⋃ij′=j+1 V (Hj′)

does not contain any inner vertex of Hj (i.e., the inner vertices of Hj

have not been touched by any of the copies Hj+1, . . . ,Hi). The firststep H0 is always fully-open by definition, and all its vertices are inner.Finally, we denote by reg(Si), deg(Si) and fo(Si) the number of regular,degenerate and fully-open steps in Si.

It follows from the definition that a newly added regular step Hi is fully-open in Si. Next, we show a series of claims which will be used later inthe proof of Claim 4.15.

Claim 4.16. Let H be a strictly `-balanced `-graph with at least threeedges. Furthermore, let G be an arbitrary `-graph and e ∈ E(G) an edgein G. Let He be an H-copy such that G ∩He = (e, e).


Then all H-copies H in G+ := G ∪ He which are not contained in Ghave the form

H = He − e+ e :=((V (He) \ e) ∪ e, (E(He) \ e) ∪ e

),

where e ∈ E(G) and |e ∩ e| > γ(H), cf. Figure 4.1.

G

e

e

He − e+ e

Figure 4.1: The possible copies of H created in a regular step. The solidlines represent He, the dashed ones H.

Proof. Let H be some H-copy in G+ which is not fully contained in G.If H = He, then the lemma is true for e = e, so we assume H 6= He.

Let e be an arbitrary edge of H which is not contained in E(He). Notethat this implies e ∈ E(G).

First we show that E(H)\e must be contained in E(He)\e, whichimplies that the two sets are equal. Assume this is not true. Set Hnew :=H[V (He)], Hold := H[V (G)] and H+e

new = Hnew +e. As we assumed thatE(H)\e * E(He)\e we know that H must contain an edge differentfrom e that is not contained in E(He) \ e, and is thus contained inE(G). This implies that e(Hold) ≥ 2. As H is not fully contained in Git must contain at least one edge of E(He)\E(G), which in turn impliesthat e(H+e

new) ≥ 2. Subgraph Hold is a strict subgraph of H as H is notfully contained in G. Moreover, H+e

new is also a strict subgraph of H asby definition E(H+e

new) ⊆ E(He) and |E(H+enew)| < |E(He)|.

One easily checks that regardless of whether e is an edge of Hnew or notwe have

e(H) = e(Hold) + e(H+enew)− 1 and v(H) ≥ v(Hold) + v(H+e

new)− `.


Thus

m`(H) =e(H)− 1

v(H)− `≤ e(Hold)− 1 + e(H+e

new)− 1

v(Hold)− `+ v(H+enew)− `

< m`(H),

which is a contradiction, as H is an H-copy. (Here the last inequalityfollows from the fact that H is strictly `-balanced and H+e

new, Hold ( Hare copies of a proper subgraph of H, each with at least `+ 1 vertices.) Hence, our assumption E(H) \ e 6= E(He) \ e is not valid.It remains to show that |e ∩ e| > γ(H). Let X := e \ e and assume|X| ≥ `−γ(H), i.e. |e∩e| ≤ γ(H). As H \e = He \e we know thatno edge of H, except e, can contain a vertex in X. Let H := H \X. Bythe previous observation we have

v(H) = v(H)− |X| ≥ `+ 1 and e(H) = e(H)− 1 ≥ 2,

thus

m`(H) ≥ e(H)− 1

v(H)− `=

e(H)− 1− 1

v(H)− |X| − `

≥ e(H)− 1− 1

v(H)− `− (`− γ(H)), (4.6)

where the last inequality holds because of the assumption on X. Wehave m`(H) = e(H)−1

v(H)−` by the assumption on H being strictly `-balanced

and m`(H) ≥ 1`−γ(H) by the definition of γ(H). Inequality (4.6) thus

implies that m`(H) ≥ m`(H), which is a contradiction as H is a copyof a proper subgraph of H with more than one edge. Thus we have|e ∩ e| > γ(H), as desired.

Note that Claim 4.16 implies that for `-graphsH with γ(H) = `−1 (andin particular for graphs) we have that G+ = G ∪ He does not containany H-copy that intersects both He \ e and G \ e. For these `-graphsthe following claim is thus straightforward while for all other `-graphsit needs a small argument.

Claim 4.17. Let 1 ≤ j ≤ i and Hj be a fully-open step in Si. Letej ∈ E(Hj) denote the attachment edge of Hj. Then any two distinctedges e, e′ ∈ E(Hj) \ ej of Hj are H-equivalent in Gi.


Proof. As Hj is fully-open in Si we know that Gi can be partitioned asGi = Hj ∪G′i such that Hj ∩G′i = ej . From Claim 4.16 we know thatany H-copy in Gi which contains some edge e ∈ E(Hj) \ ej must alsocontain all other edges e′ ∈ E(Hj) \ ej, hence the claim follows.

For i ≥ 1, let ∆(i) denote the number of fully-open copies ‘destroyed’by step Hi, i.e. let

∆(i) = |j < i | Hj fully-open in Si−1 but not in Si|.

Claim 4.18.

∆(i) ≤

1, if Hi is a regular stepv(H)− `+ 1, if Hi is a degenerate step.

Proof. Fix any edge e ∈ E(Gi−1) and let Ht, t < i, be a step withe ∈ E(Ht). Note that such a step has to exist as e ∈ E(Gi−1). Assumee contains an inner vertex of some step Hj , j < i, which is fully-open inSi−1. If t > j then Ht contains an inner vertex of Hj , which contradictsour assumption that Hj is fully-open in Si−1. If t < j then someinner vertex of Hj is contained in an edge of Ht, which contradicts thedefinition of inner vertices of Hj . It follows that t = j and e ∈ E(Hj).

This easily implies the first part of the claim. Indeed, let Hi be aregular step and ei = Hi∩Gi−1 its attachment edge. From the previousobservation we have that ei can contain inner vertices of at most oneH-copy Hj which is fully-open in Si−1, thus ∆(i) ≤ 1 as required.

Next, similarly as in the case of edges we show that any vertex v ∈V (Gi−1) can be an inner vertex of at most one H-copy Hj which isfully-open in Si−1. Fix any vertex v ∈ V (Gi−1) and assume that Hj

is fully-open in Si−1 with v being its inner vertex. Let Ht, t < i, be astep containing v. Then, by the same argument as above, it can not bethat t < j. By the definition of fully-open, the set ∪i−1

j′=j+1V (H ′j) doesnot contain any inner vertex of Hj . In particular, if t > j then this alsoholds for Ht. Therefore, v can be an inner vertex only of step Hj .

We can now derive the second part of the claim. Let Hi be a degeneratestep and e ∈ E(Hi∩Gi−1) an arbitrary edge of Hi which exists in Gi−1.By the first observation we have that e contains inner vertices of atmost one fully-open step in Si−1. By the second observation, every


vertex v ∈ V (Hi ∩Gi−1) \ V (e) is an inner vertex of at most one fully-open step in Si−1. In total, the step Hi can touch inner vertices of atmost v(H)− `+ 1 fully-open copies.

Claim 4.19. Let Hi and Hi+1 be consecutive regular steps. If ∆(i) = 1then ∆(i+ 1) = 0.

Proof. As ∆(i) = 1 we know that Hi is the first step which intersectsthe inner vertices of a fully-open step Hj in Si−1, for some j < i. Denotethe attachment edges of Hj and Hi by ej and ei, respectively. Beforestep Hi, by Claim 4.17 (if γ(H) < `−1) and Claim 4.16 (if γ(H) = `−1)the step Hj had e(H)− 1 ≥ 2 edges which were not closed in Gi−1. Weshow below that step Hi closes exactly one edge of Hj . Thus, after thestep Hi the copy Hj still contains at least one edge that is not closed.Therefore, in the (i+1)-iteration of the Algorithm 1, Hi+1 will be chosenin such a way that it intersects one of the edges of Hj which are not yetclosed. As Hi+1 is regular, it follows from the same arguments as in theproof of Claim 4.17 that it does not intersect the inner vertices of anyother fully-open step in Si and we can conclude that ∆(i+ 1) = 0.

It remains to show that Hi closes exactly one edge in Hj . We do this bya case distinction based on γ(H). Assume first that γ(H) = ` − 1 andconsider some edge e ∈ E(Hj)\ei, ej. By Claim 4.16 the only H-copyin Gi−1 that contains e is Hj . Moreover, again by Claim 4.16 the onlyH-copy in Gi which does not belong to Gi−1 is Hi. Since e /∈ E(Hi), ealso belongs to less than two copies in Gi and thus it remains not closed.

Assume now that γ(H) < ` − 1. Since in this case H contains at least4 edges, let us consider any two distinct edges e′, e′′ ∈ E(Hj) \ ei, ej.First, it follows from Claim 4.17 that e′ ≡H e′′ in Gi−1. Furthermore,let us assume that there exists an H-copy H ′ in Gi, not fully containedin Gi−1, which contains e′. Then, by Claim 4.16 there exists a uniquesuch copy H ′ = Hi − ei + e′ and |ei ∩ e′| > γ(H). However, as ei ande′ both belong to the copy Hj , this contradicts the definition of γ(H).Therefore, such an H-copy H ′ does not exist and, by symmetry, thesame is true for the edge e′′. In other words, the property that an H-copy H in Gi contains e′ if and only if it contains e′′ remains true, thuse′ is not closed in Gi.

As a final step before proving Claim 4.15, we prove a lower bound on thenumber of fully-open steps that must be contained in any grow sequenceof length s with at most d degenerate steps. Using Claim 4.19, the proof


of the following claim is identical to the proof of Claim 11 from [NS16].We include it for the sake of completeness.

Claim 4.20. For all 1 ≤ i ≤ s it holds that

fo(Si) ≥ reg(Si)/2− deg(Si) · v(H). (4.7)

Proof. Let us denote by ϕ(i) := reg(Si)/2 − deg(Si) · v(H) the righthand side of Equation (4.7). We use induction to prove the followingslightly stronger statement,

fo(Si) ≥

ϕ(i) if Hi is a regular stepϕ(i) + 1 if Hi is a degenerate step,

for all 1 ≤ i ≤ s. One easily checks that this holds for i = 1: ifH1 is a regular step then fo(S1) = 1 > 1/2, otherwise fo(S1) = 0 >−v(H) + 1. Consider now some i ≥ 2. If Hi is a degenerate step thenfrom Claim 4.18 we have ∆(i) ≤ v(H) − ` + 1 ≤ v(H) − 1 and sofo(Si) = fo(Si−1) −∆(i) ≥ fo(Si−1) − v(H) + 1. The claim now easilyfollows from reg(Si) = reg(Si−1) and deg(Si) = deg(Si−1) + 1.

Otherwise, assume that Hi is a regular step and let

j := max1 ≤ j < i | ∆(j) > 0 or Hj is a degenerate step.

Note that j is well defined, as ∆(1) = 1. Further, by the definition ofj, Hi′ is a regular step for all j < i′ ≤ i, thus ϕ(i) = ϕ(j) + (i − j)/2.In addition, we deduce from ∆(i′) = 0 for j < i′ < i that all steps Hi′

are fully-open in Si−1. We thus have

fo(Si) = fo(Sj) + (1−∆(i)) + (i− j − 1) = fo(Sj) + i− j −∆(i).

If Hj is a degenerate step then the induction assumption implies

fo(Sj) ≥ ϕ(j) + 1.

As Hi is a regular step and thus ∆(i) ≤ 1, this implies fo(Si) ≥ ϕ(j) +i − j ≥ ϕ(i), as claimed. Finally, assume that Hj is a regular copy. If∆(i) = 0, then the claim follows trivially by the induction. Otherwisewe have ∆(i) = 1 and as ∆(j) = 1 by Claim 4.19 we have that i ≥ j+2.Therefore

fo(Si) = fo(Sj) + i− j−1 ≥ fo(Sj) + (i− j)/2 ≥ ϕ(j) + (i− j)/2 = ϕ(i),

similarly as before. This finishes the proof of the claim.

4.3. Applications 51

Finally, we are ready to prove Claim 4.15.

Proof of Claim 4.15. We prove part (a) first. Let us assume that S =(H0, . . . ,Hs) contains at most d degenerate steps. Every H-copy inB := ∪si=0Hi is closed by the property of S, thus by Claim 4.17 thereare no fully-open steps in S. By Claim 4.20 this implies that

deg(S) · v(H) ≥ reg(S)/2 (4.8)

must hold. We have deg(S) ≤ d and reg(S) ≥ s − d (the first step isneither degenerate nor regular). We obtain

d · v(H) ≥ (s− d)/2.

Solving for s we get

s ≤ 2d (v(H) + 1/2) ≤ 3d · v(H),

which proves the first part.

For part (b) of Claim 4.15 let Si be a prefix of S, for some 1 ≤ i ≤ s,with at most d degenerate steps. Note that before any regular stepHj , j ≤ i created in line 11, all H-copies of Gj−1 are closed and thusby Claim 4.17 we have fo(Sj−1) = 0. Similarly as above, we havedeg(Sj−1)v(H) ≥ reg(Sj−1)/2. As we know that deg(Sj−1) ≤ d weobtain j − 1 ≤ 3dv(H), which concludes the proof.

4.3 Applications

4.3.1 Anti-Ramsey property – proper colourings

The key ingredient for the proof of Theorem 4.6 is the following lemmawhose proof we defer to the next section.

Lemma 4.21. Let H be a graph isomorphic to either a cycle on atleast 7 vertices or a complete graph on at least 19 vertices. Then forany graph G such that m(G) ≤ m2(H) it holds that G

a-ram−−−−→prp

H.

Proof of Theorem 4.6. Let H be some graph as stated in the theoremand c a constant given by Corollary 4.13 when applied to H. Let p ≤


1 G← G2 col← 0

3 while ∃e1, e2 ∈ E(G) : e1 ≡H e2 in G and e1 ∩ e2 = ∅ do4 color e1, e2 with col5 G← G \ e1, e2 and col← col + 1

6 end7 while ∃e ∈ E(G) : e does not belong to an H-copy do8 color e with col9 G← G \ e and col← col + 1

10 end11 Remove isolated vertices in G12 B1, . . . , Bk ← H-blocks obtained by applying Lemma 4.14 on G13 Colour (properly) each Bj without a rainbow H-copy using distinct

sets of colours (cf. text why this is possible)

Algorithm 2: Proper colouring without rainbow H-copy.

cn−1/m2(H) and G ∼ G(n, p). We use Algorithm 2 to find a propercolouring of G without a rainbow H-copy.

To see the correctness of the algorithm, observe first that it suffices toargue that the graph G obtained in line 11 can be properly colouredwithout a rainbow copy of H. Indeed, we only remove edges that arenot contained in an H-copy (and can thus be coloured arbitrarily) orpairs of (non-adjacent) edges that are both contained in exactly thesame H-copies (and can thus not be contained in a rainbow copy, if wegive them the same colour).

It thus remains to prove that line 13 is indeed possible. We first showthat the graph G is H-closed. Assume otherwise. Then there has toexist an H-copy H ′ which has at most two closed edges (as there areno vertices and edges which are not a part of an H-copy). If H ′ ∼= K`

then as ` ≥ 19 there at least(`2

)− 2 > ` edges of E(H ′) which are

not closed. One easily checks that this implies that there are two edgese1, e2 ∈ E(H ′) that satisfy e1 ∩ e2 = ∅ and are not closed. Thus, H ′is the only H-copy to which e1 and e2 belong, implying that e1 ≡H e2.However, this can’t be, as such a pair would have been removed in line 5of the algorithm. If H ′ ∼= C` then there are at least ` − 2 ≥ 5 edges ofH ′ which are not closed and as H ′ is a cycle two of those must be non-intersecting, again yielding a contradiction similarly as in the previous


case.

So we know that G is H-closed. We thus can apply Lemma 4.12 todeduce that we have w.h.p. that each H-block B in G satisfies m(B) ≤m2(H). By Lemma 4.14, colouring one block Bi does not influence thecolouring of any H-copy which does not lie in Bi and all Bi’s are edge-disjoint. Finally, by Lemma 4.21 there exists a desired proper colouringof every block Bi, which gives a proper colouring of G (and of the graphG) without a rainbow H-copy.

Proof of Lemma 4.21

We start with a technical observation that will help us prove the caseof forbidden complete graphs.

Claim 4.22. Let ` ≥ 4 be an integer and let G be a graph with m(G) ≤(`+1)/2. Then for any vertex v ∈ V (G) and a subset A ⊆ NG(v) of size|A| ≤ `+1, there exist at most b6·`/(`−3)c vertices w ∈ V (G)\(A∪v)with the property that G[A′ ∪ v, w] ∼= K` for some A′ ⊆ A.

Proof. First, note that if G[A] does not contain a copy of K`−2 thenthere is no such vertex w ∈ V (G)\ (A∪v). Therefore, we can assumethat |A| ≥ `− 2 and G[A] contains at least

(`−2

2

)edges. Note that then

e(G[A ∪ v]) ≥(`−2

2

)+ ` − 2. Assume now that there are k vertices

W = w1, . . . , wk ⊆ V (G) \ (A ∪ v) with the described property.Then each such vertex wi has at least `− 1 neighbours among verticesin A ∪ v, thus

e(G[A ∪ v ∪W ]) ≥ (`− 2)(`− 3)

2+ `− 2 + k · (`− 1)

=(`− 2)(`− 1)

2+ k(`− 1) = (`− 1)(`/2− 1 + k).

(4.9)

On the other hand, from m(G) ≤ (`+ 1)/2 and |A| ≤ `+ 1 we have

e(G[A∪ v ∪W ]) ≤ `+ 1

2(`+ 2 + k) = (`+ 1)(`/2 + 1 + k/2). (4.10)

Finally, combining (4.9) and (4.10) gives k ≤ 6·`/(`−3) which concludesthe proof of the claim as k has to be an integer.


Proof of Lemma 4.21 - complete graphs. Let ` ≥ 19 and G be a graphon n vertices with m(G) ≤ m2(K`) = (`+ 1)/2. It is easy to show thatthere exists an ordering v1, . . . , vn of the vertices of G such that

|N(vi) ∩ v1, . . . , vi−1| ≤ `+ 1 (4.11)

for every i ∈ [n] and let Gi := G[v1, . . . , vi]. Given a (partial) edge-colouring p of G, we say that an edge e ∈ E(G) is i-new if no edge inGi−1 is coloured with p(e). We will inductively find a proper colouringpi of Gi such that the following holds,

(i) Gi does not contain a rainbow copy of K` under colouring pi,

(ii) for every j ∈ [i]: all but at most three edges incident to vj in Gjare j-new, and

(iii) for every j < r ≤ i: if an edge vj , vr ∈ E(G) is not j-new,then there exists a subset of vertices S ⊆ v1, . . . , vj−1 such thatG[vj , vr ∪ S] ∼= K`.

The base of the induction trivially holds, thus assume that the inductionhypothesis holds for all i < k, for some 2 ≤ k ≤ n.Let pk−1 be any colouring of Gk−1 which satisfies (i)-(iii). We create acolouring pk by extending the colouring pk−1 to the edges incident to vkin Gk. Note that this implies that the only K`-copies we have to takecare of are those which contain the vertex vk. Similarly, the only edgeswhich might violate properties (ii) and (iii) are those incident to vk.

Let vi1 , . . . , viq be the neighbours of vk in Gk, with ij < ij+1 for allj ∈ [q − 1]. It follows from (4.11) that q ≤ ` + 1. Initially, assign anarbitrary new colour to each edge vk, vij for j ≤ minq, `− 2. Notethat this leaves at most three edges of Gk uncoloured, thus the property(ii) is guaranteed to be satisfied. If q < ` − 1, then the vertex vk doesnot belong to any copy of K` in Gk and properties (i) and (iii) remainsatisfied as well – in which case we are done. Therefore, from now onwe assume that q ∈ `− 1, `, `+ 1.Let R = vi`−1

, . . . , viq be the set of the remaining neighbours of vk,i.e. endpoints of edges that are not yet coloured. We first ‘clean’ Ras follows: for any vj ∈ R for which there does not exist a subsetS ⊆ v1, . . . , vj−1 such that G[S ∪ vj , vk] ∼= K`, assign an arbitrarynew colour to vj , vk and set R := R \ vj. Note that if R = ∅ after


this procedure, then vk does not belong to a copy of K` in Gk and itis easy to see that properties (i)-(iii) are satisfied. Therefore, we canassume that R 6= ∅ and observe that any colouring we assign to theremaining edges will satisfy (iii). Furthermore, note that every copy ofK` which contains vk in Gk also contains at least one vertex from R.

Before we proceed with the colouring of the remaining edges, we firstmake an observation about the colouring of the edges in Gk−1. Letvj ∈ R be an arbitrary vertex. An application of Claim 4.22 to A :=N(vj)∩v1, . . . , vj−1, which is by (4.11) at most `+1, yields that thereexist at most

b6`/(`− 3)c(`≥19)

≤ 7 (4.12)

vertices vz ∈ V (G) \ (A ∪ vj) such that there exists Sz ⊆ A withG[vj , vz∪Sz] ∼= K`. Since, by the definition of R, vk is such a vertex,it follows from (4.12) and the property (iii) that there are at most6 vertices vz, j < z < k such that the edge vj , vz is not j-new.Combining this observation with property (ii), we have that there areat most 9 colours assigned to edges incident to vj which are also assignedto some edge in Gj−1. Let us denote the set of such colours with Cjand

|Cj | ≤ 9. (4.13)

With this observation at hand, we go back to the colouring of the re-maining edges.

Let W := vi1 , . . . , vi`−2. Our aim now is as follows: for each ver-

tex vj ∈ R we want to find pairwise disjoint 2-sets Sj ⊆ W suchthat either Sj /∈ E(G) or pk−1(Sj) /∈ Cj and pk−1(Sj) 6= pk−1(Sj′)for distinct vj , vj′ ∈ R. Then the colouring can be completed by settingpk(vk, vj) := pk−1(Sj) if Sj ∈ E(G) and assigning an arbitrary newcolour otherwise. Clearly, a rainbow K`-copy which contains vk andvj ∈ R cannot contain both vertices in Sj , thus if it contains vk then ithas to miss at least |R| vertices from W ∪R. As |W | ≤ `− 2 this showsthat no such rainbow K`-copy exists, which finishes the proof.

We find these sets Sj in a greedy fashion as follows. Let R′ := Rand W ′ := W and repeat the following until R′ = ∅: if there exist twovertices a, b ∈W ′ such that a and b do not form an edge, choose vj ∈ R′arbitrarily and set Sj := a, b, R′ := R′ \ vj and W ′ := W ′ \ a, b.Otherwise, choose vj ∈ R′ arbitrarily and let a, b ∈ W ′ be such thatpk−1(a, b) /∈ Cj and pk−1(Sj) 6= pk−1(Sj′) for previously defined setsSj′ . If this procedure exhausts R′, then by the construction of the sets


Sj we are done. Furthermore, since in each iteration the size of R′decreases, it suffices to show that both cases are well-defined.

If there exists two vertices a, b ∈ W ′ that do not form an edge thenthere is nothing to show. Therefore, we can assume that W ′ induces aclique. Note that, for each vj ∈ R, at most 11 colours are forbidden;at most two because of the previously defined sets Sj′ and at most 9because of Cj . Thus, in order to show that we can find an edge Sj inW ′ which satisfies the desired property, it suffices to show that thereare more than 11 different colours appearing in the clique W ′. Since|R| ≤ 3 and |W | = ` − 2 we have |W ′| ≥ ` − 2 − 2 · 2 = ` − 6 as longas R′ 6= ∅. On the other hand, every proper colouring of a clique on atleast `− 6 vertices contains at least `− 7 > 11 different colours, whichfinishes the proof.

We remark that more careful counting of the number of different coloursin the cliqueW ′ gives a slightly better lower bound on `. Next, we provethe case of cycles.

Proof of Lemma 4.21 - cycles. Let ` ≥ 7 and G be a graph on n verticessuch that m(G) ≤ m2(C`) = 1 + 1/(` − 2). Let us assume towards acontradiction that G is a minimal graph with respect to the number ofvertices such that G a-ram−−−→

prpC`.

First, observe that in G no two vertices of degree 2 are adjacent. Tosee this, let us assume that two such vertices v1, v2 ∈ V (G) exist. ThenN(v1) ∩ N(v2) = ∅ as otherwise v1 and v2 do not belong to a C`-copythus contradicting the minimality of G. Therefore, the edges e1 and e2

incident to v1 and v2, different from the edge v1, v2, satisfy e1∩e2 = ∅.Furthermore, it follows again from the minimality of G that

G \ v1, v2a-ram−−−−→prp

C`.

Consider an arbitrary colouring of G \ v1, v2 without a rainbow C`-copy. We assign the same (new) colour to e1 and e2. Observe that norainbow C`-copy can contain both v1 and v2. On the other hand, sincee1 ≡C` e2 in G and there is no rainbow C`-copy in G \ v1, v2 thisimplies G

a-ram−−−−→prp

C`, a contradiction.

Next, it is easy to see that G does not contain a vertex v of degree 1as such a vertex does not belong to a C`-copy and would contradict theminimality of G. Therefore, δ(G) ≥ 2 and by the previous observation


the set V2 ⊆ V (G) of all the vertices of degree 2 is an independent set.We estimate the size of V2 as follows,

2m(G)n ≥ 2e(G) =∑v∈G

deg(v) ≥∑v∈V2

2 +∑

v∈V (G)\V2

3

= 2|V2|+ 3(n− |V2|)

and therefore |V2| ≥ (1 − 2/(` − 2))n. Since V2 is an independent setthis implies

(1 + 1`−2 )n ≥ e(G) ≥ e(V2, V (G) \ V2) = |V2| · 2 ≥ (2− 4

`−2 )n.

One easily checks that this a contradiction for all ` ≥ 8. For ` = 7we have that the left hand side is equal to the right hand side, whichimplies that the graph G is bipartite. Since C7 is not bipartite, Gdoes not contain C`-copy, implying the desired contradiction also inthis case.

4.3.2 Anti-Ramsey property – 2-bounded colourings

Here we give a proof of Theorem 4.4. We use the following three lemmaswhich provide a density condition of graphs that are not anti-Ramseycorresponding to the three cases from Theorem 4.4. We defer the proofsto the next subsection.

Lemma 4.23. Let H be a strictly 2-balanced graph on at least 4 verticeswhich contains a cycle and is not isomorphic to C4. Then for any graphG such that m(G) ≤ m2(H) it holds that G

a-ram−−−−→2

H.

Lemma 4.24. For any graph G such that m(G) < m2(C4) it holds thatG

a-ram−−−−→2

C4. Moreover, there exists a graph G with m(G) = m2(C4)

such that G a-ram−−−→2 C4.

Lemma 4.25. Let r, ` ∈ N be such that 2 ≤ ` ≤ r − 1 and (`, r) /∈(2, 3), (3, 4). Then for any `-graph G with m(G) ≤ m`(K

(`)r ) it holds

that Ga-ram−−−−→2 K

(`)r .

Proof of Theorem 4.4. Let ` ≥ 2 be an integer and consider a strictly `-balanced `-graph H which satisfies one of the conditions of the theoremand let c be a constant given by Corollary 4.13 when applied to H. Let


1 G← G2 col← 0

3 while ∃e1, e2 ∈ E(G) : e1 ≡H e2 in G do4 colour e1, e2 with col5 G← G \ e1, e2 and col← col + 1

6 end7 while ∃e ∈ E(G) : e does not belong to an H-copy do8 colour e with col9 G← G \ e and col← col + 1

10 end11 Remove isolated vertices in G12 B1, . . . , Bk ← H-blocks obtained by applying Lemma 4.14 with G13 Colour (2-bounded) each Bi without a rainbow H-copy using a

distinct set of coloursAlgorithm 3: 2-bounded colouring of G without rainbow H-copy.

G ∼ G(`)(n, p) for p which we will specify later. We use Algorithm 3 tofind a 2-bounded colouring of G without a rainbow H-copy.

The only difference between Algorithm 2 and Algorithm 3 is in the con-dition in line 3. In particular, in Algorithm 3 we don’t require edges e1

and e2 to be disjoint. Following the same lines as in the proof of Theo-rem 4.6 together with Lemma 4.23 (provided p ≤ cn−1/m2(H)), Lemma4.24 (provided H ∼= C4 and p n−1/m2(H)) and Lemma 4.25 (providedp ≤ cn−1/m`(H)) shows that w.h.p. G is such that the Algorithm 3 findsthe desired colouring.

Proof of Lemmas 4.23 and 4.24

Proof of Lemma 4.23 splits into a couple of cases. We first state claimswhich cover these cases. Throughout this section, we say that a 2-bounded colouring of edges incident to some vertex v is maximal if allbut at most one colour appears exactly twice.

Claim 4.26. Let G and H be graphs such that m(G) < δ(H) − 1/2.Then G

a-ram−−−−→2 H.

Proof. Consider some graphH and assume towards a contradiction that


there exists a graph G on n vertices with m(G) < δ(H)− 1/2 such thatG

a-ram−−−→2

H. Furthermore, let us assume that G is a minimal such graphwith respect to the number of vertices. It then follows from∑

v∈V (G)

deg(v) = 2e(G) ≤ 2m(G)n

that there exists a vertex u ∈ V (G) with deg(u) ≤ 2m(G) < 2δ(H)− 1.Since deg(u) ∈ Z we can further improve this bound to deg(u) ≤b2m(G)c ≤ 2(δ(H)−1). Now consider an arbitrary maximal 2-boundedcolouring of the edges incident to u and colour G− u using the min-imality assumption. Then in any rainbow subgraph of G the vertexu has degree at most δ(H) − 1, thus u cannot belong to a rainbowH-copy. However, as there are no rainbow H-copies in G − u wehave a 2-bounded colouring of G without a rainbow H-copy, which is acontradiction.

The proof of the next claim uses similar ideas as the proof of Lemma4.21 in the case of cycles.

Claim 4.27. Let G and H be graphs such that m(G) < δ(H) − 2/7,δ(H) ≥ 2 and H does not contain two adjacent vertices of degree δ(H).Then G

a-ram−−−−→2

H.

Proof. Let us consider some graph H as in the statement of the claimand assume towards a contradiction that there exists a graph G on nvertices with m(G) < δ(H) − 2/7 such that G a-ram−−−→

2H. Furthermore,

assume that G is a minimal such graph with respect to the number ofvertices.

First, we can assume that δ(G) ≥ 2δ(H) − 1 as otherwise the claimfollows from the same arguments as in the proof of Claim 4.26. Fur-thermore, similarly as in the proof of the cycle case of Lemma 4.21 wecan show that G does not contain two adjacent vertices v1, v2 ∈ V (G)with deg(v1) = deg(v2) = 2δ(H) − 1. Indeed, assume that two suchvertices v1, v2 ∈ V (G) exist. Then we colour G \ v1, v2 by the min-imality assumption without a rainbow H-copy, assign a new colour tothe edge v1, v2 and colour the remaining edges incident to v1 and v2

both by a maximal 2-bounded colouring. Then the degree of v1 andv2 in any rainbow subgraph R ⊆ G is at most δ(H). If v1, v2 ⊆ Rthen R H since H does not contain two adjacent vertices of degree


δ(H). Otherwise, v1 and v2 can have degree at most δ(H) − 1 in R,which again implies that R H or v1, v2 6∈ R. Therefore, any rainbowH-copy has to lie completely in G− v1, v2 which is not possible.

To summarize, we have δ(G) ≥ 2δ(H) − 1 and the set S ⊆ V (G) ofall the vertices of degree exactly 2δ(H) − 1 is an independent set. Weestimate the number of edges in G as follows,

2m(G)n ≥∑

v∈V (G)

deg(v)

≥ |S|(2δ(H)− 1) + (n− |S|)2δ(H) = n · 2δ(H)− |S|

and thus |S| ≥ 2n(δ(H)−m(G)). Now m(G) < δ(H)−2/7 implies that|S| > 4/7 · n. Since S is an independent set, we further have

(δ(H)− 2/7)n ≥ m(G)n ≥ e(G) ≥ e(S, V (G) \ S) ≥|S| · (2δ(H)− 1) > n(8/7 · δ(H)− 4/7),

which easily implies δ(H) < 2, hence a contradiction. Therefore, suchgraph a G does not exist.

Claim 4.28. Let H and G be graphs such that

(i) dm(G)/2e < m(H) or

(ii) dm(G)/2e = m(H), m(G) < dm(G)e and dm(G)e is odd.

Then Ga-ram−−−−→

2H.

Proof. Let H and G be graphs which satisfy condition (i) of the claim.By Lemma 2.7 there exists an orientation of the edges of G such thateach vertex has out-degree at most dm(G)e. Let us consider one such ori-entation and arbitrarily pair the out-edges incident to each vertex. As-signing the same colour to edges in each pair, in any rainbow (oriented)subgraph R ⊆ G we have for the out-degree of any vertex v ∈ V (R)

deg+R(v) ≤

⌈dm(G)e

2

⌉=

⌈m(G)

2

⌉< m(H). (4.14)

In particular, the density of R is strictly smaller thanm(H) thus R H.


Let nowH andG be graphs such that condition (ii) holds. As in the pre-vious case, let us fix an orientation of the edges of G such that each ver-tex has out-degree at most dm(G)e. Note that in every (oriented) sub-graph G′ ⊆ G there exists a vertex with out-degree strictly smaller thandm(G)e as otherwise we would have that the density of such a subgraphis dm(G)e > m(G). Therefore, we can greedily arrange the vertices ofG into a sequence v1, . . . , vn such that Ni := N+(vi)∩ vi+1, . . . , vn isof size at most dm(G)e−1. Now the colouring strategy is as follows: foreach vertex vi, first arbitrarily pair all the out-edges corresponding toNi and then all other out-edges incident to vi and assign a new colourto each pair. It remains to prove that there are no rainbow H-copiesunder such colouring.

Consider some rainbow subgraph R ⊆ G. It follows from the pairingstrategy that every vertex in R has out-degree at most ddm(G)e/2e =dm(G)/2e = m(H). Now consider the vertex vi ∈ V (R) with the small-est index i among all the vertices in R. Observe that all out-neighboursof vi in R have index larger than i. Since |Ni| ≤ dm(G)e−1 the pairingstrategy ensures that the out-degree of vi in R is at most⌈

dm(G)e − 1

2

⌉<

⌈dm(G)e

2

⌉= m(H),

where the strict inequality follows from the fact that dm(G)e is odd.Thus all the vertices in R have out-degree at most m(H) and at leastone vertex has out-degree strictly smaller than m(H). Therefore, thedensity of any rainbow subgraph R is strictly smaller than m(H) hencethere is no rainbow H-copy in G.

It remains to cover the case H = K4.

Lemma 4.29. Let G be a graph such that m(G) ≤ m2(K4) = 2.5. ThenG

a-ram−−−−→2

K4.

Proof. Let us assume towards a contradiction that there exists a graphG on n vertices with m(G) ≤ 2.5 and such that G a-ram−−−→

2K4. Without

loss of generality let G be a minimal such graph with respect to thenumber of vertices.

First, observe that G does not contain a vertex v ∈ V (G) with deg(v) <5. Otherwise, by taking any maximal colouring of edges incident to v,we have that no rainbow K4-copy can contain v. Since it follows from


the minimality of G that there is no rainbow K4-copy in G \ v weget that G does not contain a rainbow H-copy, thus a contradiction.Therefore, δ(G) ≥ 5 and since∑

v∈V (G)

deg(v) ≤ m(G) · 2n ≤ m2(K4) · 2n = 5n

it follows that G is 5-regular. Observe that G K6, as the colouring(see Figure 4.2)

(v1, v2, v1, v3), (v1, v4, v1, v5),(v1, v6, v5, v6), (v2, v4, v2, v6),(v3, v4, v3, v6), (v3, v5, v2, v5), (v4, v5, v4, v6)

shows that K6a-ram−−−−→2 K4.

Figure 4.2: A colouring of K6 without a rainbow K4-copy.

Let now v ∈ G be an arbitrary vertex andN(v) = w1, . . . , w5. Assumefirst that δ(G[N(v)]) ≤ 2 and w.l.o.g. let w1, w2 and w3 be the verticessuch that w1, w2, w1, w3 /∈ E(G). Consider the following colouringof the edges incident to v:

(v, w1), (v, w2, v, w3), (v, w4, v, w5).

Now any possible rainbowK4-copy which contains the vertex v must alsocontain the vertex w1 and one of the vertices from w2, w3. However,that is not possible as w1 is not connected to any of w2 and w3. On theother hand, by the minimality of G no rainbow K4-copy lies completelyinG\v. ThusG contains no rainbowK4-copy, which is a contradictionwith the choice of G.


Therefore, we can assume that δ(G[N(v)]) ≥ 3. As G is 5-regular,this implies that every vertex wi ∈ N(v) has at most one neighbour inV (G) \ (N(v) ∪ v). Thus, any K4-copy that contains a vertex fromN(v) ∪ v can contain at most one vertex from V (G) \ (N(v) ∪ v),which in turn implies that any such clique has to contain three verticesin N(v). However, one easily checks that this can only be if one of theremaining two vertices in N(v) has degree at most two within G[N(v)],which we have already excluded. Thus, there exists no K4-copy whichcontains a vertex in N(v) and a vertex in V (G) \ (N(v)∪ v). We canthus colour G[N(v) ∪ v] and G[V (G) \ (N(v) ∪ v)] separately andby the minimality of G a colouring without rainbow K4-copy exists forboth these graphs. This concludes the proof of the lemma.

We are now ready to combine the previous claims.

Proof of Lemma 4.23. Let us first consider a graph H on four vertices.There exist only two such graphs that are strictly 2-balanced: C4 andK4. Therefore, if H is a graph on four vertices then H ∼= K4 and theconclusion of the lemma follows from Lemma 4.29. For the rest of theproof we assume that H contains at least 5 vertices and since H is astrictly 2-balanced graph we have δ(H) ≥ 2.

Let m2(H) = k + x for some k ∈ N, k ≥ 1 and x ∈ [0, 1). Observe thatδ(H) > m2(H) as otherwise removing a vertex with degree at mostm2(H) would result in a graph with the same or larger 2-density, whichcannot be since H is strictly 2-balanced. Thus δ(H) ≥ k+1. If x < 1/2then m(G) < k+1/2 = (k+1)−1/2 and the lemma follows from Claim4.26. So we may assume in the following that x ≥ 1/2.

One easily checks that

3

4v(H)2 − v(H) >

(v(H)

2

)≥ e(H)

(as v(H) ≥ 5) and thus

e(H)

v(H)+ 3/2 >

e(H)− 1

v(H)− 2.

As x ≥ 1/2 this implies m(H) > m2(H) − 3/2 ≥ k − 1. For k ≥ 3 wetherefore have

dm(G)/2e ≤ d(k + 1)/2e(k≥3)

≤ k − 1 < m(H),


and Ga-ram−−−−→

2H follows from Claim 4.28. So from now on we may assume

that x ≥ 1/2 and k ∈ 1, 2.Furthermore, if H contains two adjacent vertices of degree δ(H) thenfrom the fact that H is strictly 2-balanced and v(H) ≥ 5 we have

e(H)− 1− (2δ(H)− 1)

v(H)− 2− 2<e(H)− 1

v(H)− 2

and so (2δ(H)−1)/2 > m2(H) ≥ k+1/2. Therefore, either δ(H) ≥ k+2or δ(H) = k+ 1 and H does not contain two adjacent vertices of degreeδ(H). In the first case we trivially have m(G) ≤ m2(H) < k + 1 <k+ 2− 1/2 and the lemma follows again from Claim 4.26. In the lattercase, if we additionally assume that x < 5/7 then

δ(H)− 2/7 ≥ k + 5/7 > k + x = m2(H)

and the lemma follows from Claim 4.27. Thus we may assume from nowon that x ≥ 5/7 and k ∈ 1, 2.Finally, if e(H) < (5v(H)2 − 3v(H))/14 then

e(H)

v(H)+

5

7>e(H)− 1

v(H)− 2,

and x ≥ 5/7 implies that m(H) > m2(H)−5/7 ≥ k. Similarly as beforewe have

dm(G)/2e ≤ d(k + 1)/2e ≤ k < m(H),

for k ∈ 1, 2 and the lemma follows from Claim 4.28.

To summarize, we have shown that Ga-ram−−−−→2 H unless the following three

conditions hold simultaneously:

(a) x ≥ 5/7,

(b) k ∈ 1, 2 and

(c) e(H) ≥ (5v(H)2 − 3v(H))/14.

Let us consider some H such that all three properties apply. Then from(b) and (c) we have

3 > m2(H) ≥ (5v(H)2 − 3v(H))/14− 1

v(H)− 2. (4.15)


A simple calculation yields that (4.15) implies v(H) < 7. If v(H) = 6then from (c) we have e(H) ≥ 12 while from m2(H) < 3 we obtaine(H) ≤ 12. But then m(H) ≥ 2 and dm2(H)e = 3 and the lemmafollows from the part (ii) of Claim 4.28. Otherwise, if v(H) = 5 thenfrom (c) we have e(H) ≥ 8 while from m2(H) < 3 we obtain e(H) ≤ 9.However, for e(H) ∈ 8, 9 we have m2(H) ∈ 2 + 1/3, 2 + 2/3 thus Hdoes not satisfy (a). This finishes the proof.

Proof of Lemma 4.24. Let us assume, towards a contradiction, that G isa graph on n vertices such that m(G) < m2(C4) = 3/2 and G a-ram−−−→

2C4.

Furthermore, let G be a minimal such graph with respect to the numberof vertices. Then ∑

v∈V (G)

deg(v) ≤ 2m(G)n < 3n

implies that there exists a vertex v ∈ V (G) with deg(v) ≤ 2. ColouringG \ u by the minimality assumption on G and the two edges incidentto v with the same (new) colour yields a colouring of G with no rainbowC4-copy, contradicting our choice of G.

Figure 4.3: A counter-example for the case H = C4.

For the second part of the lemma, consider the graph C3+6 given in

Figure 4.3. It is easy to see that m(C3+6 ) = 3/2. Furthermore, it follows

from the fact that the graph is 3-regular that every pair of edges iscontained in at most two C4-copies. As there are 9 edges in C3+

6 , inevery 2-bounded colouring there are at most 4 pairs of edges which arecoloured the same. It now follows from the previous observation thatevery such pair of edges can prevent at most two C4-copies from being


rainbow. However, there are 9 copies of C4, thus at least one copy hasto be rainbow. This finishes the proof.

Proof of Lemma 4.25

We use the following notion of a link in a hypergraph.

Definition 4.30 (Hypergraph link). Let ` ≥ 2 be an integer and G an`-graph. Then for a vertex v ∈ V (G) we define the link of v in G to bethe (`− 1)-graph Gv induced by the set of edges

e \ v : e ∈ E(G) and v ∈ e .

Furthermore, define the link of two vertices v, w in G to be the (`− 2)-graph Gv,w induced by the set of edges

e \ v, w : e ∈ E(G) and v, w ∈ e .

We make a series of claims towards the proof of Lemma 4.25.

Claim 4.31. Let G be a vertex minimal `-graph such that G a-ram−−−→2 K(`)r .

ThenGu

a-ram−−−→2

K(`−1)r−1 .

for every vertex u.

Proof. Assume the contrary. Then there exists a 2-bounded colouringcu of Gu without a rainbow K

(`−1)r−1 -copy. Let c be the partial colouring

of G given byc(e) := cu(e \ u)

for all e ∈ E(G) with u ∈ e. Then u cannot belong to a rainbowK

(`)r -copy in G. As we can also colour G\u without a rainbow K

(`)`+1-

copy by the the minimality assumption on G, this thus contradicts theassumption of the claim G

a-ram−−−→2 K(`)r .

Claim 4.32. Let G be a graph with at most 8 edges. Then G a-ram−−−→2

K3

if and only if G contains a copy of K4. Furthermore, if G[K] ∼= K4 forsome K ⊆ V (G), then for every T ∈

(K3

)there is a 2-bounded colouring

of G with G[T ] being the only rainbow K3-copy in G.


Proof. One easily checks that K4a-ram−−−→

2K3, thus if G contains K4

then Ga-ram−−−→2 K3 as well. In the other direction, let G be a vertex

minimal graph with at most 8 edges without a copy of K4 such thatG

a-ram−−−→2

K3. If v(G) ≥ 6 then δ(G) ≤ b16/6c = 2, allowing thus a 2-bounded colouring without a rainbow K3-copy similar to the argumentin Lemma 4.24. Otherwise, for v(G) ∈ 4, 5 one easily checks thatG

a-ram−−−−→2 K3, thus contradicting the choice of G.

For the furthermore-part, observe that if G[K] ∼= K4 for some K ⊆V (G), then G contains at most two additional edges e1, e2 /∈ G[K]. Letus assume that K = v1, v2, v3, v4 and, without loss of generality, T =v1, v2, v3. Then the following 2-bounded colouring has the requiredproperty:

(e1, e2), (v1, v2, v1, v3), (v1, v4, v4, v2), (v2, v3, v3, v4).

Claim 4.33. Let G be a 3-graph with at most 16 edges and no isolatedvertices. Then G a-ram−−−→

2K

(3)4 if and only if G is isomorphic to a 3-graph

which consists of two copies of K(3)5 that share 4 vertices.

Proof. If G consists of two copies of K(3)5 that share 4 vertices, then

v(G) = 6, e(G) = 16 and G contains 9 copies of K(3)4 . Since any pair

of edges coloured the same can prevent at most one rainbow K(3)4 -copy

and in any 2-bounded colouring of G there are at most 8 different pairsof edges which are coloured the same, it follows that one copy of K(3)

4

will always be rainbow.

In the other direction, let G be a vertex-minimal 3-graph on the vertexset v1, . . . , vn with at most 16 edges such that G a-ram−−−→

2K

(3)4 . If n ≤ 5

then G ⊆ K(3)5 and the following 2-bounded colouring of K(3)

5 gives acontradiction with the choice of G:

(v1, v2, v5, v1, v3, v5), (v1, v4, v5, v3, v4, v5),(v2, v4, v5, v1, v2, v4), (v2, v3, v4, v2, v3, v5),(v1, v2, v3, v1, v3, v4).

Therefore, from now on we can assume that n ≥ 6. Next, let us assumetowards the contradiction that G does not contain a K(3)

5 -copy. Let vi


be a vertex of minimum degree which is at most b16 · 3/nc ≤ 48/6 = 8.Then by Claims 4.31 and 4.32 and the minimality assumption on G, thelink of vi contains a K4-copy. As G does not contain a K(3)

5 -copy, weknow that there exists a 3-subset T of the vertices of a K4-copy in Gvsuch that T /∈ E(G). Since e(Gv) ≤ 8, Claim 4.32 asserts the existenceof a 2-bounded colouring cv of Gv such that the only rainbow K3-copyis induced by T . By the minimality of G we can colour G \ v withouta rainbow K

(3)4 -copy. We then extend such colouring to G by using cv

to colour the edges containing v, thus obtaining a colouring without arainbow K

(3)4 -copy. This is a contradiction with the choice of G.

Without loss of generality, we may now assume G[v1, . . . , v5] ∼= K(3)5 .

Then from e(G) ≤ 16 and e(K(3)5 ) = 10 it follows that degG(vi) ≤ 6 for

every vi ∈ v6, . . . , vn. By Claims 4.31 and 4.32, we know that the linkof every vertex has to contain a copy of K4. Thus Gv6

∼= K4 and everyedge of G has to either contain v6 or belong to G[v1, . . . v5]. This isonly possible if v(G) = 6 and so G is isomorphic to two copies of K(3)

5

that share 4 vertices.

We combine the previous claims to derive the following lemma, whichwe then use as a base for the induction in the proof of Lemma 4.25.

Lemma 4.34. If G is a 4-graph with m(G) ≤ 4 then Ga-ram−−−−→

2K

(4)5 .

Proof. Suppose the claim is false and let G be a vertex-minimal 4-graphwith m(G) ≤ 4 and G a-ram−−−→

2K

(4)5 . Since

4 ≥ m(G) ≥∑

x∈V (G)

deg(x)/(4v(G)),

it follows from the minimality of G and Claims 4.31 and 4.33 that forall x ∈ V (G) we have deg(x) = 16 and the link Gx is isomorphic totwo copies of K(3)

5 sharing 4 vertices. Consider any vertex x ∈ V (G)

and let two copies of K(3)5 in Gx be on the vertex sets a1, b1, b2, b3, b4

and a2, b1, b2, b3, b4. Note that x, a1, a2, bi /∈ E(G) for every bi ∈b1, b2, b3, b4.Next, we consider the link Ga1

. Then b1, b2, b3, b4, x ∈ V (Ga1) and let

a′ be the remaining vertex. If a′ 6= a2 then there exists bi, say b1, such


that b2, b3, b4, a1, a2, a′, x ∈ V (Gb1), which is not possible. Applying

the same argument to Ga2, we have

V (Ga1) = b1, b2, b3, b4, x, a2 and V (Ga2

) = b1, b2, b3, b4, x, a1.

It follows now from x, a1, a2, bi /∈ E(G) that b1, b2, b3, b4 induces aK

(3)4 -copy in Ga1 and Ga2 , and furthermore b1, bj , a1, a2 ∈ E(G) for

every bj ∈ b2, b3, b4. This implies deg(b1) ≥ 18, thus a contradiction.

We are now ready to prove Lemma 4.25. We split the proof into twoparts. First, we consider cliques of the type K(`)

`+1.

Proof of Lemma 4.25 – small cliques K(`)`+1, ` ≥ 4 . We prove the asser-

tion by induction on `. The case ` = 4 follows from Lemma 4.34 asm4(K

(4)5 ) = 4. Next, let ` > 4 and assume that the claim holds for

K(`−1)` . Let us assume towards the contradiction that there exists an

`-graph G with m(G) ≤ m`(K(`)`+1) = ` such that G a-ram−−−→2 K

(`)`+1. Fur-

thermore, let G be a vertex-minimal such `-graph. Claim 4.31 implies

Gua-ram−−−→2 K

(`−1)`

for every vertex u ∈ V (G). By the induction hypothesis we must have

m(Gu) > m`−1(K(`−1)` ) = `− 1.

Consider some S ⊆ V (Gu) such that m(Gu) = e(Gu[S])/|S|. Note that|S| ≥ ` + 2 as otherwise e(Gu[S]) ≤

(`+1`−1

)=(`+1

2

)and thus m(Gu) ≤

`/2 < `− 1, contradicting our assumption. Hence, e(Gu) ≥ e(Gu[S]) =m(Gu) · |S| ≥ (` − 1)(` + 2) > `2. On the other hand, a vertex u ofminimum degree satisfies

e(Gu) ≤ ` ·m(G) ≤ ` ·m`(K(`)`+1) = `2, (4.16)

yielding the desired contradiction.

Proof of Lemma 4.25 – r ≥ `+ 2. The proof goes by induction on `.For ` = 2 the claim follows from Lemma 4.23. Let now ` > 2 andassume that the claim holds for all K(`−1)

r with r ≥ `+ 2.


Let us assume towards a contradiction that there exists some r ≥ ` +2 and an `-graph G with m(G) ≤ m`(K

`r) such that G a-ram−−−→

2K

(`)r .

Furthermore, we assume that G is a minimal such `-graph with respectto the number of vertices. We show that then

δ(G) > (r + 1) ·m`−1(K(`−1)r−1 ). (4.17)

Assuming that equation (4.17) holds, we can lower bound m(G) as fol-lows,

e(G)/v(G) =

∑v∈V (G) deg(v)

v(G) · `>r + 1

`·m`−1(K

(`−1)r−1 )

=(r + 1) · (

(r−1`−1

)− 1)

`(r − 1− `+ 1)=

(r + 1)(r−1`−1

)`(r − `)

− r + 1

`(r − `)(r≥`+2)

≥(r + 1) · `r

(r`

)`(r − `)

− r + 1

r>r + 1

r·m`(K

(`)r )− r + 1

r

= m`(K(`)r ) +

m`(K(`)r )− (r + 1)

r. (4.18)

On the other hand, for r ≥ `+ 3 and since ` ≥ 3 we have

m`(K(`)r ) =

(r`

)− 1

r − `≥(r3

)− 1

r − 3≥ r + 1.

Furthermore, for r = ` + 2 ≥ 6 we have m`(K(`)r ) ≥ r + 1 as well.

Together with (4.18) this implies m(G) > m`(K(`)r ) for r ≥ ` + 2 but

(r, `) 6= (5, 3), which contradicts our choice of G in this case. It remainsto consider the cases r = 5 and ` = 3. One easily checks that in thiscase

m(G)(4.18)>

r + 1

`·m`−1(K

(`−1)r−1 ) ≥ m`(K

(`)r ),

again contradicting the assumption on G. Therefore, no such G existsand the claim follows.

It remains to prove equation (4.17). Consider some vertex u ∈ V (G) ofminimum degree. Similarly to the case of K(`)

`+1 cliques, the minimalityof G implies that

Gua-ram−−−→2 K

(`−1)r−1 . (4.19)

With (4.19) it follows from the induction assumption that

m(Gu) > m`−1(K(`−1)r−1 ). (4.20)


One easily checks that

m(K(`−1)r ) =

1

r

(r

`− 1

)=

1

r· r

r − `+ 1

(r − 1

`− 1

)<

(r−1`−1

)− 1

r − `= m`(K

(`−1)r−1 ).

Together with (4.20) this implies that the densest subgraph of Gu hasto be a graph on at least r + 1 vertices. Thus, we get from (4.20) thate(Gu) > (r+ 1) ·m`−1(K

(`−1)r−1 ) and as δ(G) ≥ e(Gu) this concludes the

proof of (4.17).

4.3.3 Ramsey property for graphs and hypergraphcliques

As the last application of our method we give a proof of the 0-statementof Theorem 3.1 (except for some special cases) and Theorem 4.2.

Proof of Theorem 4.2. Observe that if a hypergraph G is not 2-boundedanti-Ramsey for H then it is also not Ramsey for H. Indeed, considersome 2-bounded colouring of G without a rainbow copy of H. As eachcolour occurs at most twice, we can colour one edge red and the other oneblue. Now observe that any monochromatic subgraph in this colouringcorresponds to a rainbow subgraph in the original colouring. Thus, nomonochromatic copy of H appears.

As an immediate consequence of Theorem 4.4 we get the 0-statement ofTheorem 3.1 in the case the 2-densest subgraph is not a triangle. Un-fortunately, the case when the 2-densest subgraph is a triangle requirescertain modifications of the H-closed property thus we do not stateit here (see [NS16] for the whole proof). For the case of hypergraphcliques, we immediately obtain Theorem 4.2 for all `-graphs which arecliques of size at least `+ 1 with the exception of the (hyper)graphs K3

and K(3)4 . The case of K3 follows from Theorem 3.1, thus it remains to

consider K(3)4 .

Note that Algorithm 3, with line 4 changed such that it assigns redcolour to e1 and blue to e2, provides a 2-colouring of the hypergraphG. As the analysis and the correctness of the algorithm remains thesame as in the proof of Theorem 4.4, it suffices to show that if m(G′) ≤m3(K

(3)4 ) = 3 then G′ → K

(3)4 .


Let G be a vertex minimal graph with G ram−−→2 K(3)4 and let u ∈ V (G) be

a vertex of minimum degree. Claim 4.31 yields Guram−−→

2K3. However,

deg(u) ≤ 3 · m(G) ≤ 3 · m3(K(3)4 ) = 9 and it is easy to see that any

graph with less than 15 edges is not Ramsey for K3 and two colours,see e.g. [EFRS78].

Chapter 5Universality of random graphs

5.1 Introduction

Given a family of graphs H, a graph G is said to be H-universal if itcontains every member of H as a subgraph (not necessarily induced).Universal graphs have been studied quite extensively, particularly withrespect to families of forests, planar graphs and graphs of bounded de-gree (see, for example, [AC08, ACK+00, BCE+82, BCS11, BCLR89,CG83, DKRR15, FNP16, KL14] and their references). In particular, itis of interest to find sparse universal graphs.

Let H∆(n) be the family of all graphs on at most n vertices with max-imum degree at most ∆. Building on earlier work with several authors[AA02, ACK+00, ACK+01], Alon and Capalbo [AC07, AC08] showedthat there are graphs with at most c∆n2−2/∆ edges which are Hn(∆)-

73

74 Chapter 5. Universality of random graphs

universal. A simple counting argument shows that this result is bestpossible.

The construction of Alon and Capalbo is explicit. An earlier approachhad been to study whether random graphs could be H∆(n)-universal.The first result on universality in random graphs was proved by Alon,Capalbo, Kohayakawa, Rödl, Ruciński and Szemerédi [ACK+00], whoshowed that for any ε > 0 and any natural number ∆ there exists a con-stant C > 0 such that the random graph G(n, p) is a.a.s. H∆((1− ε)n)-universal for p ≥ (C log n/n)1/∆. In this theorem, some slack is al-lowed by only asking that the random graph contains subgraphs with(1− ε)n vertices (the so-called almost-spanning universality). However,one can also ask whether the random graph G(n, p) contains all sub-graphs of maximum degree ∆ with exactly n vertices, that is, whetherit is H∆(n)-universal. Because we no longer have any extra room to ma-noeuvre, this problem is substantially more difficult to treat than thealmost-spanning version. Nevertheless, Dellamonica, Kohayakawa, Rödland Ruciński [DKRR15] have shown that for any natural number ∆ ≥ 3there exists a constant C > 0 such that G(n, p) is a.a.s. H∆(n)-universalfor p ≥ (C log n/n)1/∆. The case ∆ = 2 was later treated by Kim andLee [KL14], who obtained similar bounds to those in [DKRR15].

Results on embedding large bounded-degree graphs in the random graphhave proved useful in other contexts [BKT13, KRSS11]. For example, akey component in the proof that the size Ramsey number of bounded-dedgree graphs is subquadratic [KRSS11] is an embedding lemma likethat used in [ACK+00]. In the next chapter we use ideas developed hereto further improve the bound on size Ramsey numbers of such graphs.

We make some initial progress on these problems by improving thetheorem of Alon, Capalbo, Kohayakawa, Rödl, Ruciński and Szemerédi[ACK+00] on almost-spanning universality as follows.

Theorem 5.1. For any ε > 0 and integer ∆ ≥ 3 there exists C > 0such that if

p ≥(C log2 n

n

)1/(∆−1)

.

then the random graph G(n, p) is a.a.s. H∆((1− ε)n)-universal.

This result bypasses a natural barrier, since (log n/n)1/∆ is roughly thelowest probability at which we can expect that every collection of ∆vertices will have many neighbours in common, a condition which is

5.1. Introduction 75

extremely useful if one wishes to embed graphs of maximum degree ∆.On the other hand, the lowest probability at which one might hope thatthe random graph G(n, p) is a.a.s. H∆((1− ε)n)-universal is n−2/(∆+1).Indeed, below this probability, G(n, p) will typically not contain (1 −ε) n

∆+1 vertex-disjoint copies ofK∆+1 (see, for example, [JŁR11]). Thus,for ∆ = 3 our result is optimal up to the logarithmic factor, while for∆ ≥ 4 the gap remains.

The proof of Theorem 5.1 presented here is based on [CFNŠ16]. How-ever, we will make use of the result on universality for degenerategraphs (Theorem 3.6) proved in Chapter 3 rather than the the re-sult from [FNP16], as it was done in [CFNŠ16]. This gives us some-what easier proof and a better bound on p. When embedding a graphH ∈ H∆((1 − ε)n), we will first find a (∆ − 1)-degenerate subgraphH ′ ⊆ H by removing all small components and certain short cycles inH. We then embed H ′, after which we place the deleted pieces in anappropriate way so as to obtain an embedding of H.

5.1.1 Overview of the proof

Before we give an overview of our proof, it is instructive to first discusswhy is (log n/n)1/∆ a “natural” bound for the universality property.Consider a graph H ∈ H∆((1− ε)n) and let G ∼ G(n, p). We say thatan injective function f : V (H) → V (G) is an embedding of H into Gif f(u), f(w) ∈ E(G) for every u,w ∈ E(H). Suppose we wish tofind such a function iteratively, vertex by vertex. Let h1, . . . , h` be anordering of the vertices of H and for each i = 1, . . . , ` choose f(hi) asfollows:

(i) let Ni be the set of the candidates for f(hi) with respect to leftneighbours of hi, i.e.

Ni := V (G) ∩⋂j<i

hi,hj∈E(H)

NG(f(hj));

(ii) if Fi := Ni \ f(h1), . . . , f(hi−1) is non-empty then set f(hi) :=vi for some vi ∈ Fi and otherwise terminate the procedure withfailure.

If the procedure does not terminate with failure then the obtained map-ping is clearly an embedding of H into G. Let us assume now that H is


∆-regular. Then regardless of the ordering we inevitably arrive in thesituation where all ∆ neighbours of some vertex hi have already beenembedded. Taking p to be asymptotically smaller than (log n/n)1/∆,the threshold for the property that every subset of ∆ vertices have anon-empty common neighbourhood, likely results in Ni = ∅ and theprocedure terminates with a failure. Note that Ni 6= ∅ does not implyFi 6= ∅, but it is a necessary condition.

It turns out, to our knowledge, that almost all results on embeddings oflarge (general) graphs either essentially follow the described procedure(e.g. see [ABH+13, ABET15, AF92, DKRR15]) and therefore requirea lower bound on p of order (log n/n)1/∆, or are entirely based on thesecond-moment method (e.g. see [Rio00]). However, even though thesecond-moment method proofs might give better bounds on p, it seemsdifficult to make use of them for proving universality in random graphsand finding large subgraphs in random-like graphs.

We overcome this bottleneck by combining the two approaches. Themain idea is to separate a graph H ∈ H∆((1− ε)n) into two parts: onewhich is (∆−1)-degenerate and the other which is composed of cycles oflength at most 2 log n. By ordering the vertices of the first part such thateach vertex hi has at most ∆−1 neighbours in h1, . . . , hi−1, from theearlier discussion we have that p ≥ (log n/n)1/(∆−1) suffices to guaranteeNi 6= ∅. However, since H is a large graph there is still an issue of freespace, that is, we need Fi 6= ∅ as well. By having a slightly larger p, weshowed in Chapter 3 that it can indeed be done. On the other hand,using Janson’s inequality to embed each cycle from the second part atonce, rather than vertex by vertex, turns out to give enough leverage topush the bound on p significantly below (log n/n)1/∆. Note that the keyfeature of this idea is that the Janson’s inequality is used only to showthe existence of relatively small structures, thus giving enough room toovercome all necessary union-bound calculations.

5.2 Universality for small graphs

In this section, we prove auxiliary lemmas which will allow us to ignoresmall components in the proof of Theorem 5.1 (see Phase III in theproof of Theorem 5.1).

Lemma 5.2. Let ∆ ≥ 3 and k be integers and let H ∈ H∆(logk n) be aconnected graph with v(H) ≥ ∆ + 2. If p ≥ n−2/(∆+1) then G ∼ G(n, p)

5.2. Universality for small graphs 77

contains H with probability 1− e−ω(n).

Proof. Let (h1, . . . , hv(H)) be an arbitrary ordering of the vertices of Hand let V1, . . . , Vv(H) ⊆ V (G) be disjoint subsets of order n/ logk n. Wewish to use Janson’s inequality to prove the lemma. For that, we willrestrict our attention to the “canonical" copies of H: the family Hii∈Iconsists of all those copies of H in Kn with the property that the vertexhj belongs to the set Vj for every j ∈ 1, . . . , v(H). We now estimatethe parameters µ and δ defined in Theorem 2.2.

Let X be the number of copies Hi, i ∈ I, that appear in G. Thenµ = E[X] satisfies the bound

µ =

(n

logk n

)v(H)

pe(H) ≥(

n

logk n

)v(H)

pv(H)∆/2

≥(

n

logk n· n−∆/(∆+1)

)v(H)

n,

where the last inequality follows from v(H) ≥ ∆ + 2.

Next, note that for any proper subgraph J ⊂ H we have

• e(J) ≤(v(J)

2

)if v(J) ≤ ∆, and

• e(J) ≤ 12 ((v(J)− 1)∆ + ∆− 1) = 1

2 (v(J)∆− 1) otherwise.

The second estimate follows from the fact that H is connected andtherefore there exists at least one vertex in J with degree at most ∆−1.We can now rewrite δ as

δ =∑

(i,j)∈I×IHi∼Hj

pe(Hi)+e(Hj)−e(Hi∩Hj) =∑J⊂H

∑Hi∩Hj∼=J

p2e(H)−e(J).

Using the observations above, we split the sum based on the size of v(J),getting

δ ≤∑J⊂Hv(J)≤∆

∑Hi∩Hj∼=J

p2e(H)−(v(J)2 )+

∑J⊂H

v(J)≥∆+1

∑Hi∩Hj∼=J

p2e(H)−(v(J)∆−1)/2.

We can bound the number of summands by first deciding on an embed-ding of J , which can be done in (n/ logk n)v(J) ways, and then on an


embedding of the remaining parts of the two copies of H which intersecton J (at most (n/ logk n)2(v(H)−v(J)) ways), yielding

δ ≤∆∑j=2

(v(H)

j

)(n

logk n

)2v(H)−j

p2e(H)−(j2)

+

v(H)−1∑j=∆+1

(v(H)

j

)(n

logk n

)2v(H)−j

p2e(H)−(j∆−1)/2.

Finally, by pulling µ2 outside, we obtain

δ ≤ µ2

(∆∑j=2

(v(H)

j

)(n

logk n

)−jp−(j2)

+

v(H)−1∑j=∆+1

(v(H)

j

)(n

logk n

)−jp−(j∆−1)/2

).

By substituting for p, we get the following upper bound on the firstsum,

δ1 :=

∆∑j=2

(v(H)

j

)(n

logk n

)−jp−(j2) ≤

∆∑j=2

(log n)2kjn−jnj(j−1)∆+1

≤∆∑j=2

(log n)2∆kn∆

∆+1 (j−1)−j = (log n)2∆k∆∑j=2

n−∆+j∆+1

≤ (log n)2∆k∆n−1−1/(∆+1) 1/n.

Proceeding similarly for the second sum, we have

δ2 :=

v(H)−1∑j=∆+1

(v(H)

j

)(n

logk n

)−jp−(j∆−1)/2

≤v(H)−1∑j=∆+1

(log n)2jkn−jnj∆−1∆+1 =

v(H)−1∑j=∆+1

(log n)2jkn−j+1∆+1 .

Since j ≥ ∆ + 1 in the above sum, we have

n−j+1∆+1 = n−1− j−∆

∆+1 (log n)−2k(j+1)n−1,

5.2. Universality for small graphs 79

thus it easily follows that δ2 = o(1/n). Summing up, we get δ = o(µ2/n).

Finally, by applying Theorem 2.2 with parameters µ and δ, we obtain

Pr[X < µ/2] ≤ e−µ2/(8(µ+δ)).

Since µ n, this implies the conclusion of the lemma.

The next lemma deals with graphs on at most ∆ + 1 vertices. Since wetreat ∆ as a constant, it is a standard application of Janson’s inequalityand we omit the proof.

Lemma 5.3. Let ∆ ≥ 3 be an integer and H any graph on at most∆ + 1 vertices. If p n−2/(∆+1) then G ∼ G(n, p) contains H withprobability 1− e−ω(n).

Finally, we make use of Lemmas 5.2 and 5.3 to show that every largeinduced subgraph of G(n, p) contains all connected graphs from thefamily H∆(logk n), simultaneously.

Lemma 5.4. Let ε > 0 be a constant and ∆ ≥ 3 and k be integers.Then, for p n−2/(∆+1), G ∼ G(n, p) a.a.s. has the following property:for every V ′ ⊆ V (G) of order |V ′| ≥ εn, G[V ′] contains every connectedgraph H ∈ H∆(logk n).

Proof. Let G ∼ G(n, p), with p as stated in the lemma. By Lemmas 5.2and 5.3, for a fixed subset V ′ ⊆ V (G) of order |V ′| ≥ εn and a connectedgraph H ∈ H∆(logk+1(εn)), we have

Pr[H ⊆ G[V ′]] = 1− e−ω(n).

Note that as logk n ≤ logk+1(εn), these estimates also apply for everyconnected graph H ∈ H∆(logk n). Since there are at most 2n choicesfor V ′ and at most

logk n∑vH=2

∆vH/2∑eH=1

(v2H

eH

)≤ ∆ log2k n ·

(log2k n

∆ logk n

)= o(2n)

connected graphs H ∈ H∆(logk n), an application of the union boundcompletes the proof.


5.3 Disjoint representatives in hypergraphs

The following lemma will allow us to place a set of short cycles in theproof of Theorem 5.1 (see Phase II in the proof of Theorem 5.1). Thebound on parameter t and the size of the set D are chosen according tothe application.

Lemma 5.5. Let ε > 0 be a constant and ∆ ≥ 3 be a (fixed) integer,let 3 ≤ g ≤ 2 log n and t ≤ εn/(32 log2 n) be integers and let D ⊆ [n] bea subset of size εn/(4 log n). Then there exists a constant C = C(ε) > 0such that if

p ≥(C log3 n

n

)1/(∆−1)

then G ∼ G(n, p) satisfies the following with probability at least 1 −o(1/n): for any family of subsets Wi,j(i,j)∈[t]×[g], where

(i) Wi,j ⊆ V (G) \D and |Wi,j | = ∆− 2 for all (i, j) ∈ [t]× [g], and

(ii) Wi,j ∩Wi′,j′ = ∅ for all i 6= i′,

there exists a family of cycles Ci = (ci1 , . . . , cig )i∈[t], each of length g,such that

(i) V (Ci) ⊆ G[D] and V (Ci) ∩ V (Ci′) = ∅, for all i 6= i′, and

(ii) Wi,j ⊆ NG(cij ) for all (i, j) ∈ [t]× [g].

Lemma 5.5 will follow as an application of the following generalizationof Hall’s matching criterion of Haxell [Hax95] and Lemma 5.7. Given afamily E of subsets of some ground set V , we denote with τ(E) the sizeof a smallest subset X ⊆ V such that E ∩X 6= ∅ for every E ∈ E .

Theorem 5.6 (Theorem 3, [Hax95]). Let r ∈ N and Hi = (V, Ei)i∈Ibe a family of r-uniform hypergraphs on the same vertex set. If

τ(⋃i∈I′Ei) > (2r − 1)(|I ′| − 1)

for every I ′ ⊆ I, then there exists a family of hyperedges eii∈I suchthat ei ⊆ Ei and ei ∩ ej = ∅ for every i 6= j ∈ I.

5.3. Disjoint representatives in hypergraphs 81

Note that the statement of Theorem 3 in [Hax95] is slightly differentthan Theorem 5.6, however it is easy to see that they are equivalent.

The following lemma will be used to verify conditions of Theorem 5.6applied on hypergraphs given by cycles spreading over neighbourhoodsof certain vertices.

Lemma 5.7. Let ε > 0 be a constant and ∆ ≥ 3 be a (fixed) integer, let3 ≤ g ≤ 2 log n and k ≤ εn/(32 log2 n) be integers and let D ⊆ [n] be asubset of size εn/(4 log n). Then there exists a constant K = K(ε) > 0such that if

p ≥(K log3 n

n

)1/(∆−1)

then G ∼ G(n, p) satisfies the following with probability at least 1 −o(1/n2): for any family of subsets Wi,j(i,j)∈[k]×[g], where

(i) Wi,j ⊆ V (G) \D and |Wi,j | = ∆− 2 for all (i, j) ∈ [k]× [g], and

(ii) Wi,j ∩Wi′,j′ = ∅ for all i 6= i′,

and any subset D′ ⊆ D of order |D′| ≥ |D| − 2gk, there exists i ∈1, . . . , k and a cycle C = (c1, . . . , cg) ⊆ G[D′] of length g such thatWi,j ⊆ NG(cj) for all j ∈ 1, . . . , g.

Proof. Our aim is to show that for a subset D′ ⊆ D and a fam-ily Wi,j(i,j)∈[k]×[g] satisfying properties (i) and (ii), the graph G ∼G(n, p) fails to satisfy the conclusion of the lemma with probability atmost e−Ω(k log2 n). Since we can choose the family Wi,j(i,j)∈[k]×[g] inat most

(n

∆−2

)kg ≤ 2(∆−2)kg logn ≤ 22∆k log2 n ways and D′ in at most(n

2gk

)≤ 22gk logn ≤ 24k log2 n ways, the lemma follows by a simple appli-

cation of the union bound. It remains to prove the desired bound onthe probability of a failure.

We first introduce some notation. Given a cycle C = (c1, . . . , cg) ⊆ Kn

of length g, we define the graph C ⊕ i by

V (C ⊕ i) = V (C) ∪g⋃j=1

Wi,j , and E(C ⊕ i) = E(C) ∪⋃j∈[g]v∈Wi,j

cj , v.


Furthermore, let V1, . . . , Vg ⊆ D′ be arbitrarily chosen disjoint subsetsof order εn/(8g log n) (this is possible since |D′| ≥ εn/(8 log n)), definethe family of canonical cycles C as

C := C = (c1, . . . , cg) | C is a cycle and cj ∈ Vj for all j ∈ [g]

and set

C+ := C ⊕ i | C ∈ C and i ∈ [k].

Observe that if G contains any graph from C+, then G contains thedesired cycle. Using Janson’s inequality, we upper bound the probabil-ity that this does not happen. In the remainder of the proof, we willestimate the parameters µ and δ defined in Theorem 2.2.

Note that each graph C+ ∈ C+ appears in G with probability

pg+(∆−2)g = p(∆−1)g.

Therefore, we have

µ = |C+|p(∆−1)g = k

(εn

8g log n

)gp(∆−1)g k log2 n.

Next, we wish to show that δ = o(µ2/k log2 n). By definition, we have

δ =∑i,j∈[k]

∑C′,C′′∈C

C′⊕i∼C′′⊕j

pe(C′⊕i)+e(C′′⊕j)−e((C′⊕i)∩(C′′⊕j)).

We consider the cases i 6= j and i = j separately.

First, if C ′ ⊕ i ∼ C ′′ ⊕ j for i 6= j and C ′, C ′′ ∈ C, then we have(C ′ ⊕ i) ∩ (C ′′ ⊕ j) = C ′ ∩ C ′′. Let J := C ′ ∩ C ′′ and observe thate(J) ≥ 1, as otherwise C ′ ⊕ i and C ′′ ⊕ j would not have any edges incommon. Let J1 be the family consisting of all possible graphs of theform C ′ ∩ C ′′,

J1 := J = C ′ ∩ C ′′ | C ′, C ′′ ∈ C and e(J) ≥ 1.

5.3. Disjoint representatives in hypergraphs 83

We can now estimate the contribution of such pairs to δ as follows:

δ1 =∑i 6=j

∑C′,C′′∈C

e(C′∩C′′)≥1

pe(C′⊕i)+e(C′′⊕j)−e(C′∩C′′)

=∑i 6=j

∑J∈J1

∑C′,C′′∈CC′∩C′′=J

p2(∆−1)g−e(J)

≤ k2∑J∈J1

(εn

8g log n

)2(g−v(J))

p2(∆−1)g−e(J)

= µ2∑J∈J1

(εn

8g log n

)−2v(J)

p−e(J).

Since e(J) = 1 for v(J) = 2 and e(J) ≤ v(J) otherwise, we can boundthe last sum by

∑J∈J1

(εn

8g log n

)−2v(J)

p−e(J)

≤∑J∈J1

v(J)=2

(εn

8g log n

)−4

p−1 +∑J∈J1

v(J)>2

(εn

8g log n

)−2v(J)

p−v(J).

Observe that there are at most(gj

)(εn/(8g log n))

j graphs J ∈ J1 on jvertices. Thus, we have∑

J∈J1

(εn

8g log n

)−2v(J)

p−e(J)

≤(g

2

)(εn

8g log n

)−2

p−1 +

g∑j=3

(g

j

)(εn

8g log n

)−jp−j

o(1/n) +

g∑j=3

(2 log n

j

)(8g2 log n

εn· n1/(∆−1)

)j= o(1/n) = o(1/(k log2 n)),

where we used that ∆ ≥ 3. Therefore, we obtain δ1 = o(µ2/k log2 n).

Let us now consider the case C ′⊕i ∼ C ′′⊕i, for some i ∈ [k] and distinctcycles C ′, C ′′ ∈ C. Let J := C ′ ∩ C ′′ and observe that v(J) ≥ 1 and


v(J) ≤ g − 1. As before, let J2 be the family consisting of all possiblegraphs of the form C ′ ∩ C ′′,

J2 := J = C ′ ∩ C ′′ | C ′, C ′′ ∈ C and v(J) ∈ 1, . . . , g − 1.

Note that if V (J) ∩ Vq = v for some q ∈ 1, . . . , g, then v, w ∈E(C ′ ⊕ i ∩ C ′′ ⊕ i) for all w ∈Wi,q. Therefore, we have

e(C ′ ⊕ i ∩ C ′′ ⊕ i) = e(J) + v(J)(∆− 2).

With these observations in hand, we can bound the contribution of suchpairs to δ as follows:

δ2 =∑i∈[k]

∑C′,C′′∈C

v(C′∩C′′)≥1

pe(C′⊕i)+e(C′′⊕i)−e(C′⊕i∩C′′⊕i)

=∑i∈[k]

∑J∈J2

∑C′,C′′∈CC′∩C′′=J

p2(∆−1)g−(e(J)+v(J)(∆−2))

≤ k∑J∈J2

(εn

8g log n

)2(g−v(J))

p2(∆−1)g−(e(J)+v(J)(∆−2))

≤ µ2

k

∑J∈J2

(εn

8g log n

)−2v(J)

p−(e(J)+v(J)(∆−2)).

Since J is a proper subgraph of a cycle we have e(J) ≤ v(J)−1. There-fore, we can bound the last sum by

∑J∈J2

(εn

8g log n

)−2v(J)

p−(e(J)+v(J)(∆−2))

≤∑J∈J2

(εn

8g log n

)−2v(J)

p−v(J)(∆−1)+1.

Note that there are at most(gj

)(εn/(8g log n))

j graphs J ∈ J2 with j

5.4. Proof of the main theorem 85

vertices. Thus, for sufficiently large K = K(ε) > 0 we have

∑J∈J2

(εn

8g log n

)−2v(J)

p−v(J)(∆−1)

≤g−1∑j=1

(g

j

)(εn

8g log n

)−jp−j(∆−1)+1

g−1∑j=1

(8g2 log n

εn· n

K log3 n

)jp = o(1/ log2 n),

with room to spare. Therefore, we obtain δ2 = o(µ2/(k log2 n)).

Finally, we have δ ≤ δ1 + δ2 = o(µ2/(k log2 n)) and Theorem 2.2 givesthe desired upper bound on the probability that G does not contain anygraph from C+, completing the proof of the lemma.

Proof of Lemma 5.5. Let G ∼ G(n, p) with p as stated in the lemmaand C = K5.7(ε). For each i ∈ [t], we define a g-uniform hypergraphHi := (D,Ei) as follows: a set of vertices v1, . . . , vg ⊆ D forms ahyperedge if and only if G[v1, . . . , vg] contains a cycle C = (c1, . . . , cg)such that Wi,j ⊆ NG(cj) for all j ∈ [g]. Taking a union bound over allk ∈ [t], with probability at least 1−o(1/n) (by Lemma 5.7) we have thatfor every I ⊆ [t] and D′ ⊆ D of order |D′| ≥ |D|−2g|I| the hypergraphHI =

⋃i∈I E(Hi) contains an edge which completely lies in D′. This

shows that τ(HI) ≥ 2g|I| and, by Theorem 5.6, the desired family ofvertex-disjoint cycles (i.e. hyperedges) exists.

5.4 Proof of the main theorem

Our proof strategy goes as follows. Given a graph H ∈ H∆((1 − ε)n)we first remove small connected components from H, writing H1 forthe resulting graph. Working in H1, we then remove one induced cycleof length at most 2 log n (i.e. we remove vertices) from each connectedcomponent of H1 which is not (∆− 1)-degenerate. Using Theorem 3.6we find an embedding of H2 and then, using Lemma 5.5, we place theremoved cycles into G in an appropriate way. Finally, using Lemma 5.4,


we complete the embedding of H by embedding small components oneby one.

Proof of Theorem 5.1. Let R,D3, . . . , D2 logn ⊆ [n] be arbitrarily cho-sen disjoint subsets of 1, . . . , n such that |R| = (1− ε/2)n and |Di| =εn/(4 log n) for each i ∈ 3, . . . , 2 log n. Let G be a graph with thefollowing properties:

(i) the induced subgraph G[R] is D∆((1− ε)n,∆− 1)-universal,

(ii) for every subset V ′ ⊆ V (G) of order |V ′| ≥ εn, the inducedsubgraph G[V ′] contains every connected graph from the familyH∆(log3 n), and

(iii) G satisfies the property given by Lemma 5.5 for every 3 ≤ g ≤2 log n, t ≤ εn/(32 log3 n) and D = Dg.

By Theorem 3.6 and Lemmas 5.4 and 5.5, G ∼ G(n, p) satisfies proper-ties (i)–(iii) asymptotically almost surely provided that

p ≥(C log3 n

n

)1/(∆−1)

,

for sufficiently large C. We remark that the bound on p here is deter-mined by Lemma 5.5. Next, we show that these properties imply thatG is H∆((1− ε)n)-universal.

Preparing the graph H. Let H ∈ H∆((1− ε)n) and let H1 ⊆ H bethe subgraph which consists of all connected components of H with atleast log3 n vertices. The following observation plays a crucial role inour argument.

Claim 5.8. For every connected component Q in H1 at least one of thefollowing properties hold:

(a) Q is (∆− 1)-degenerate, or

(b) Q contains a cycle of length at most 2 log n.

Proof. Let us assume that a connected component Q does not satisfy(a). Then Q is ∆-regular. Pick an arbitrary vertex v ∈ Q and observethat if Q does not contain a cycle of length at most 2 log n then the


subgraph of Q induced by vertices at distance at most log n from v isa tree. However, such tree contains more than

∑lognj=1 (∆ − 1)j > n

vertices, which cannot be. Therefore, Q contains a cycle of length atmost 2 log n.

Let us enumerate connected components ofH1 asQ1, . . . , Qk and let I ⊆[k] denote the indices of components which are not (∆− 1)-degenerate.For each i ∈ I let Ci be a smallest cycle in Qi and let ì denotes itslength. Note that such cycle is then necessarily induced and ì ≤ 2 log n,by Claim 5.8. Furthermore, fix an arbitrary ordering (c1i , . . . , c

ìi ) of the

vertices along Ci. Finally, let H2 := H1 \[⋃

i∈I V (Ci)].

Phase I: Embedding H2 into G[R]. Since from each componentQi in H1 which was not (∆ − 1)-degenerate we have removed at leastone vertex, the resulting component in H2 is then necessarily (∆ − 1)-degenerate. Therefore H2 is (∆ − 1)-degenerate and by property (i)there exists an embedding f : V (H2)→ R of H2 into G[R].

Phase II: Embedding removed cycles. Consider some 3 ≤ g ≤2 log n and let Ig ⊆ I be the set of all indices i ∈ I such that ì =

g. For each (i, j) ∈ Ig × [g], let Wi,j := f(NH(cji ) ∩ V (H2)). Notethat, by construction, the family of subsets Wi,j(i,j)∈Ig×[g] satisfiesrequirements (i) and (ii) of Lemma 5.5 with D = Dg. Moreover, sinceeach component in H1 is of size at least log3 n we have |Ig| ≤ |I| ≤n/ log3 n. Therefore, by property (iii) of the graph G (that is, by Lemma5.5), there exists a family (ci,1, . . . , ci,g)i∈Ig of vertex disjoint cyclesin G[Dg] such that setting f(cji ) := ci,j for every (i, j) ∈ Ig× [g] gives anembedding of H2∪

[⋃i∈Ig Ci

]into G[R∪Dg]. Since this holds for every

3 ≤ g ≤ 2 log n and the sets D3, . . . , D2 logn are disjoint, we obtain anembedding of H1 into G.

It is worth remarking that the bound on I, which facilitates the ap-plication of Lemma 5.5, is the reason why we treat small connectedcomponents separately.

Phase III: Embedding small components. As a last step, we haveto extend our embedding of H1 to an embedding of the whole graph H.Using the facts that H is of order (1 − ε)n and each component of Hwhich is not in H1 is of order at most log3 n, we can greedily embedthese components one by one as follows. Consider one such component


and let V ′ ⊆ V (G) be the set of vertices which are not an image of somealready embedded vertex of H. Then |V ′| ≥ εn and, by property (ii) ofthe graph G, G[V ′] contains an embedding of the required component.Repeating the same argument for each component which has not yetbeen embedded, we obtain an embedding of the graph H.

Chapter 6Size-Ramsey numbers of graphs with

bounded degree

6.1 Introduction

Given a graph H, the size Ramsey number r(H) is the smallest integerm such that there exists a graph G with m edges which is Ramseyfor H. This notion was first introduced by Erdős, Faudree, Rousseauand Schelp [EFRS78]. See Section 1.3 for summary of some importantresults.

In this chapter we are interested in size Ramsey numbers of bounded-degree graphs. The best known upper bound for graphs with maximum

89

90 Chapter 6. Size-Ramsey numbers of graphs with bounded degree

degree ∆ is due to Kohayakawa, Rödl, Schacht and Szemerédi [KRSS11],

r(H) ≤ Cn2−1/∆(log n)1/∆, (6.1)

where n is the number of vertices of H and C = C(∆) is a constant. Onthe other hand, Rödl and Szemerédi [RS00] showed that there exists agraph H with n vertices and r(H) ≥ n log1/60 n. This leaves a large gapbetween the two bounds. Here we further improve the bound in (6.1) inthe case where H is additionally triangle-free (i.e. H does not containK3 as a subgraph). This includes, among others, bipartite graphs.

Theorem 6.1. For any integer ∆ ≥ 5 there exists a constant C > 0such that for every triangle-free graph H with n vertices and maximumdegree at most ∆ we have

r(H) ≤ Cn2−1/(∆−1/2)(log n)1/(∆−1/2).

The restriction ∆ ≥ 5 in Theorem 6.1 is purely of technical nature andcomes from the proof of Lemma 6.5. We believe that it can be avoided.However, the requirement that H is triangle-free is more critical andremoving it either requires delicate technical work or, most likely, newideas. We derive Theorem 6.1 as a consequence of the stronger Theorem6.3, which we present next.

Instead of asking for a monochromatic copy of one graph H, we con-sider the stronger property of containing a monochromatic copy of everygraph from some familyH in the same colour. More precisely, a graph Gis said to be H-Ramsey-universal if every two-colouring of the edges ofG contains a monochromaticH-universal graph. Note that the notion ofRamsey-universality corresponds to the notion of partition-universalityfrom [KRSS11].

Similarly as in [KRSS11], we show that a random graph G(N, p) isalmost surely Ramsey-universal, for p which gives the existence of asparse Ramsey-universal graph with the number of edges bounded as inTheorem 6.1. The main result in [KRSS11] reads as follows:

Theorem 6.2. For every integer ∆ ≥ 2 there exist constants B ∈ Nand C > 0, such that if N = Bn and p = p(N) ≥ (C logN/N)1/∆ then

limn→∞

Pr [G(N, p) is H∆(n)-Ramsey-universal ] = 1.

Our main result is the following theorem which improves upon Theorem6.2 when restricted to the family of triangle-free graphs.

6.1. Introduction 91

Theorem 6.3. For every integer ∆ ≥ 5 there exist constants B ∈ Nand C > 0, such that if N = Bn and p = p(N) ≥ (C logN/N)

1∆−1/2

thenlimn→∞

Pr [G(N, p) is F∆(n)-Ramsey-universal ] = 1.

As already mentioned, the requirement ∆ ≥ 5 in Theorem 6.3 is anartefact of the proof. Since the number of edges of G(N, p) is almostsurely at most (1+ε)N2p, Theorem 6.3 implies the existence of a graphG with at most

(1 + ε)N2p = (1 + ε)(Bn)2

(C log(Bn)

Bn

) 1∆−1/2

= C ′n2−1/(∆−1/2)(log n)1/(∆−1/2)

edges which is F∆(n)-Ramsey-universal, for a suitable constant C ′. Thisproves Theorem 6.1.

At first, the bound on p in Theorem 6.3 might seem arbitrary. However,a careful inspection shows that for ∆ = 3 such value of p matches thethreshold for H = K4 in Theorem 3.1 (Ramsey’s theorem for randomgraphs), up to the logarithmic factor. In particular, this means thatusing the same approach to handle the case where ∆ = 3 (with trianglesallowed) cannot give better bound.

Organization. The chapter is organised as follows. In the next sectionwe give an overview of the proof of Theorem 6.3, highlighting the maindifficulties and ideas. Section 6.3 introduces the regularity method andthe main technical machinery behind the proof of Theorem 6.3. InSection 6.4, by repeating the argument of Chvátal et al. we reduce theF∆(n)-Ramsey-universality of random graphs to F∆(n)-universality ofrandom-like graphs given by the regularity method. Then, by combiningthe embedding strategy from Section 3.2, the decomposition result andthe regularity properties, we show F∆(n)-universality of such graphs.

6.1.1 Overview of the proof

Using the regularity method, Chvátal, Rödl, Szemerédi and Trotter[CRST83] reduced the problem of proving that KN is Ramsey for someH ∈ H∆(n) to proving that certain random-like graphs are universal forH∆(n) (note that there is no colouring involved in the second problem).


Following the development of the sparse regularity method and, in par-ticular, the result on inheritance of regularity on a fine scale of Gerkeet al. [GKRS07], Kohayakawa, Rödl, Schacht and Szemerédi [KRSS11]extended this approach to random graphs. Here we follow the samestrategy, with the new ingredient being the more efficient EmbeddingLemma (Lemma 6.17).

To prove the Embedding Lemma (Lemma 6.17) we apply a similarstrategy as in Chapter 5. Using Lemma 6.5, we partition vertices ofH ∈ F∆(n) into V1 and V2 such that:

1. H[V1] (the subgraph of H induced by vertices in V1) is ‘almost’(∆− 2)-degenerate and

2. each connected component of H[V2] is either a cycle of lengthat most log n or is of constant size and satisfies certain densityproperties.

The first part is then embedded one vertex at a time, and the secondpart one connected component at a time. We briefly discuss differencescomparing to Chapter 5 (refer to Section 5.1.1 for notation).

• The (∆−2)-degeneracy (instead of ∆−1) comes from the fact thatthe random-like setting we work with allows us to reason aboutedges between two sets only if both of them are of size Ω(1/p). Ifwe would have a vertex hi with |NH(hi, h1, . . . , hi−1)| = ∆− 1then we expect |Ni| np∆−1 and by having p asymptoticallybelow n−1/∆ we get |Ni| = o(1/p). Now, if there exists an edgehi, hz ∈ E(H) for some z > i, we would like to choose f(hi)such that it has a good degree into

N ′z = V (G) ∩⋂j<i

hj ,hz∈E(H)

NG(f(hj)).

In particular, if we lose the control over NG(f(hi)) ∩N ′z then wealso lose the control overNz and the process potentially terminatesat the step z. However, if Ni is rather small we cannot claim thatthere exists a choice of f(hi) which has a good degree into Nz.

• Unfortunately, we are not able to show that H[V1] is (∆ − 2)-degenerate while keeping the structure of connected components

6.2. Decomposition of triangle-free graphs 93

of the second part under control. We overcome this by showingthat there exists a choice of V2 and an ordering of V1 such thatvertices of V1 which have ∆ − 1 neighbours embedded prior totheir embedding (instead of at most ∆− 2) have a rather specialstructure. We pay for this by having more complicated structuresin H[V2] instead of just cycles. This is done in Lemma 6.2.

• In order to embed a component from H[V2] at once rather thanvertex by vertex, we apply the recently proven KŁR conjectureas the replacement for Janson’s inequality in Chapter 5. This isone of the main reasons why we fail to handle triangles: each ofthe vertices from a triangle might have ∆ − 2 already embeddedneighbours and therefore their candidate sets are of sizeO(np∆−2).However, calculation shows that if we have three sets of size np∆−2

and p = o(n−1/∆) then the KŁR conjecture cannot guarantee thatthere exists a copy of K3 touching each of them. Another issueis that the KŁR conjecture is only meaningful for constant sizesubgraphs. Fortunately, if the component we wish to embed isnot of constant size we know it is a cycle, which allows us to dealwith it without resorting to the KŁR conjecture. Lemma 6.14summarizes this.

6.2 Decomposition of triangle-free graphs

We prove that the vertex set of every graph H ∈ F∆(n) can be parti-tioned into subsets V ′ and Q such that the induced subgraph H[V ′] is“almost” (∆− 2)-degenerate and each connected component of H[Q] iseither a short cycle or satisfies certain 2-density condition. This densitymeasure is tightly related to the KŁR conjecture, which we introducein the next section. We also need the following definition.

Definition 6.4. For an integer k ∈ N and subsets U,W ⊆ V (G), wesay that U is (k,W )-independent in G if there is no path of length atmost k (i.e. a path with at most k + 1 vertices) with endpoints beingdistinct vertices in U and all other vertices being in W .

Properties of the decomposition formally read as follows.

Lemma 6.5. Let H ∈ F∆(n) for some ∆ ≥ 5. Then there exists a


partition

V (H) =

[t⋃i=1

(ri ∪ Si)

]∪Q

of the vertices of H such that the following holds for every i ∈ [t]:

(a) degH(ri, r1, . . . , ri−1) ≤ ∆− 2,

(b) Si ⊆ NH(ri),

(c) degH(s,Q) = 1 and degH(s, r1, . . . , ri) = ∆− 1 for every s ∈ Si,and

(d) ri ∪ Si is (3,⋃tj=i+1(rj ∪ Sj) ∪Q)-independent.

Moreover, for every connected component Q′ of H[Q] we have

• Q′ is a cycle of size at most 2 log n and/or

• v(Q′) ≤ 5∆2, δ(Q′) ≥ 2 and m2(Q′) ≤ 3/2.

Before we give the proof, we briefly discuss properties given by Lemma6.5. First, the condition ∆ ≥ 5 is an artefact of the proof and we believethat ∆ ≥ 3 suffices. Note that the union of ri∪Si’s represents V ′ fromthe preceding discussion. The property (a) implies that H[r1, . . . , rt]is (∆−2)-degenerate. Moreover, in foresight we have that the vertices inV ′ which have ∆−1 embedded neighbours prior to their embedding areexactly vertices in Si’s. For them the lemma provides additional struc-ture: the last neighbour of a vertex s ∈ Si into r1, . . . , rt is ri and, by(d), vertices in Si have the strong independence property. By embed-ding vertices in Si right after the vertex ri, this structural informationwill make it possible to keep the candidate sets under control. Unfor-tunately, the exact reason why this is the case (and how we do it) is ofsomewhat technical nature and will become apparent only in the proofof the Embedding Lemma (Lemma 6.17). The minimum degree and the2-density of connected components in H[Q] spring a relation betweenthe KŁR conjecture and the bound on p in Theorem 6.3. In particular,the smaller 2-density of components in H[Q] the better bound on p weget. In order to push p below n−1/∆, it turns out that anything smallerthan 2 suffices. Therefore, one of the reasons why we fail to handletriangles is m2(K3) = 2.


Proof of Lemma 6.5. Let q ∈ N0 be the largest number for which thereexists a family C = Cii∈[q] of disjoint subsets of V (H) such that eachH[Ci] is a cycle of size at most 2 log n and e(Ci, Cj) = 0 for distinct i, j ∈[q]. If there is more than one such family, choose arbitrary one whichminimizes

∑qi=1 |Ci|. Let VC =

⋃i∈[q] Ci and set Γ := NH(VC , V (H) \

VC) and L := V (H) \ (VC ∪ Γ). Observe that H[L] is 2-degenerate:otherwise, it contains a subgraph of minimum degree at least 3 and itis easy to show that such subgraph contains an induced cycle of size atmost 2 log n. This contradicts the maximality of q.

If q = 0, then L = V (H) and by 2-degeneracy there exists an orderingr1, . . . , rv(H) such that degH(ri, r1, . . . , ri−1) ≤ 2 ≤ ∆− 2, thus weare done. Therefore, we may assume q ≥ 1 (hence VC 6= ∅). We nowdefine the set Q. Let U1, . . . , Uq be a family of disjoint subsets ofΓ ∪ L which maximizes

∑i∈[q] |Ui| under the following constraints:

(aQ) e(Ci ∪ Ui, Cj ∪ Uj) = 0 for distinct i, j ∈ [q],

(bQ) if |Ci| > 5 then Ui = ∅, and

(cQ) if |Ci| ≤ 5 then the induced subgraph Qi = H[Ci ∪ Ui] satisfiesm2(Qi) ≤ 3/2, δ(Qi) ≥ 2 and distQi(u,Ci) ≤ 2 for every u ∈ Ui.

Since H is triangle-free, each cycle Ci is of length at least 4 and onecan check that m2(Ci) ≤ 3/2. Therefore, Q is well defined. Further-more, note that distHi(u,Ci) implies v(Qi) ≤ 5∆2 and the connectedcomponents induced by Q :=

⋃i∈[q](Ci ∪ Ui) satisfy requirements of

the lemma. Let VU :=⋃i∈[q] Ui and write Γ1 ⊆ Γ \ (VU ∪ NH(VU ))

for the subset of all vertices s with degH(s, VC) = 1. Similarly, letL0 := L\(VU ∪NH(VU )) and write L1 ⊆ L\VU for the subset of all ver-tices v with degH(v, VU ) = 1. We need the following two claims whichfollow from the maximality of VU :

(i) If s1, s2 ∈ Γ1 are distinct vertices such that either s1, s2 ∈E(H) or NH(s1, L0) ∩ NH(s2, L0) 6= ∅, then s1, s2 is (3, Q)-independent.

(ii) If s ∈ Γ1 and v ∈ NH(s, L1) then s, v is (3, Q)-independent.

Before we prove these claims we first finish the proof of the lemma.Let Γ+ = Γ \ (Γ1 ∪ VU ) and L+ = L \ (VU ∪ L0 ∪ L1), and note thatdegH(v,Q) ≥ 2 for every v ∈ Γ+ ∪ L+. Furthermore, we split L0 into


L+0 and L−0 , with L

+0 ⊆ L0 being the subset of all vertices v ∈ L0 with

degH(s,Γ1) ≥ 2 and L−0 := L0\L+0 . Consider a maximal subset Γ′1 ⊆ Γ1

such that there exists an ordering r1, . . . , rt of V ′ = (L\VU )∪Γ+∪Γ′1which satisfies (a) and

(aR) vertices in L−0 ∪ L1 precede vertices in L+ ∪ Γ+ ∪ L+0 ,

(bR) degH(ri, r1, . . . , ri−1 ∩ (L−0 ∪ L1)) ≤ 2 for every ri ∈ L−0 ∪ L1,and

(cR) vertices in L−0 ∪ L1 ∪ L+ ∪ Γ+ precede vertices in L+0 .

Note that for Γ′1 = ∅ such ordering can easily be obtained from the2-degeneracy of H[L−0 ∪ L1] and degH(v,Γ1 ∪ Q) ≥ 2 for every v ∈L+

0 ∪ L+ ∪ Γ+. Therefore, Γ′1 is well defined. Next, observe that S :=Γ1 \ Γ′1 is an independent set in H. Indeed, if there exists an edges1, s2 ∈ E(H[S]) then setting rt+1 := s1 contradicts the maximalityof Γ′1. The same argument shows degH(s) = ∆ for every s ∈ S. Wenow partition S into S1, . . . , St according to properties (b) and (c) of thelemma: s ∈ Si (1 ≤ i ≤ t) iff i is the largest index such that s ∈ NH(ri).By the definition, each set Si satisfies (b). Since degH(s, VU ) = 0 fors ∈ Γ1 and S is an independent set, we have degH(s, V ′) = ∆ − 1.Therefore S1, . . . , St is indeed a partition of S and (c) holds as well.It remains to verify (d). We show a stronger statement: ri ∪ Si is(3, ri+1, . . . , rt ∪ (S \ Si) ∪Q)-independent for every i ∈ [t].

First, observe that for every i ∈ [t] with Si 6= ∅ we have

degH(ri, r1, . . . , ri−1) = ∆− 2. (6.2)

Let us assume that this is not true for some ri. Set rj+1 := rj (fort ≥ j ≥ i) and ri := s for arbitrary s ∈ Si. It is easy to see that (a)remains satisfied in the new ordering. Since (aR)–(cR) are not influencedby this shifting we get a contradiction with the maximality of Γ′1. Thisimmediately gives Si = ∅ for ri ∈ L+∪Γ+. Next, consider some ri ∈ L−0and suppose that Si 6= ∅. Then Si = NH(ri,Γ1) (since vertices in L−0have at most one neighbour in Γ1 ⊇ S) and (aR) imply

NH(ri, r1, . . . , ri−1) = NH(ri, r1, . . . , ri−1 ∩ (L−0 ∪ L1)).

From (bR) we then conclude degH(ri, r1, . . . , ri−1) ≤ 2 ≤ ∆−3, whichcontradicts (6.2). Therefore, for ri ∈ L−0 ∪L+ ∪Γ+ we have Si = ∅ andthe property (d) is trivially satisfied. Next, consider ri ∈ L1 (and in


parallel ri ∈ Γ′1) and suppose Si 6= ∅. From degH(ri, Q) = 1 and (6.2)we have Si = s. Moreover, using additionally the fact that S is anindependent set, we also have

NH(ri, s, ri+1, . . . , rt ∪ (S \ s)) = ∅. (6.3)

(Recall that NH(X,Y ) stands for the union of neighbourhoods of ver-tices v ∈ X in Y , not their common neighbourhood). If ri, s is not(3, ri+1, . . . , rt ∪ (S \ s) ∪Q)-independent then (6.3) implies ri, sis not (3, Q)-independent, which contradicts claim (ii) (claim (i)). Fi-nally, consider ri ∈ L+

0 and let us assume Si 6= ∅. Note that (6.2)implies |Si| ≤ 2. If Si = s1, s2 then (6.2) and the fact that S is anindependent set further imply

NH(ri, s1, s2, ri+1, . . . , rt ∪ (S \ Si)) = ∅. (6.4)

From the second part of the claim (i) we have that s1, s2 is (3, Q)-independent, which together with (6.4) shows that it is also

(3, ri+1, . . . , rt ∪ (S \ Si) ∪Q)-independent.

Since NH(ri, Q) = ∅, we conclude that ri satisfies (d). Let us nowsuppose Si = s. Without loss of generality we may assume

v = NH(ri, ri+1, . . . , rt ∪ (S \ Si)).

Moreover, from NH(ri, Q) = ∅ we have that any path of length at most3 between ri and s with all internal vertices being in ri+1, . . . , rt ∪(S \ s) ∪ Q induces a path of length at most 2 between v and s,with the same restriction on internal vertices. From (cR) we have v ∈L+

0 ∪ Γ′1. If v ∈ L+0 , then the property (d) follows from NH(v,Q) = ∅

and NH(s, (S \ s) ∪ ri+1, . . . , rt) = ∅. Otherwise, if v ∈ Γ′1 and theproperty (d) is violated, from NH(s, (S \ s)∪ri+1, . . . , rt∪VU ) = ∅we haveNH(s, VC)∩NH(v, VC) 6= ∅. But this implies s, v is not (3, Q)-independent, which contradicts the claim (i). Therefore ri satisfies (d),which finishes the proof of the lemma.

It remains to prove claims (i) and (ii). We only give proof of the claim(i), since the claim (ii) follows by the same argument.

Proof of the claim (i). By the definition of Γ1 we have degH(sj , VC) = 1and degH(sj , VU ) = 0 (j = 1, 2). Therefore, we may assume thereis a unique Ci ∈ C which contains c1 = NH(s1, VC) and c2 =


NH(s2, VC). If s1, s2 is not (3, Q)-independent, then either c1 = c2or c1, c2 ∈ E(H). In any case, there exists a cycle s1, s2 ⊆ C ′ ⊆s1, s2 ∪ Ci ∪ L0 of size at most 5. Since such cycle has no commonedge with any other cycle Cj , j 6= i, if |Ci| > 5 then by exchangingCi and C ′ we get a contradiction with the minimality of

∑qi=1 |Ci|.

Otherwise, let us consider Q′i = H[Ci ∪ Ui ∪ s1, s2 ∪ v], where v ∈NH(s1, L0)∩NH(s2, L0) is an arbitrary element if such exists and v = ∅otherwise. It is easy to verify that Q′i satisfies (cQ), which contradictsthe maximality of VU . Therefore, s1, s2 is (3, Q)-independent.

6.3 The regularity method

In this section we introduce the machinery used to prove Theorem 6.3.Generally speaking, the Regularity Lemma imposes random-like prop-erties which are useful for finding certain subgraphs. The workflow ofthe proof of Theorem 6.3 is briefly as follows: We start with an arbi-trary colouring of a random graph G ∼ G(n, p) and select a subgraphGc induced by a suitable colour c. Since Gc might have lost certainproperties of the random graph (for example, the structure of edges be-tween large subsets), we use the Regularity Lemma to restore them upto a certain level. Finally, we use these properties to find a copy of thegraph under consideration. This approach was pioneered by Chvátal etal. [CRST83].

In the next subsection we give basic definitions and state the version ofSzemerédi’s Regularity Lemma suitable for sparse graphs. In Section6.3.2 we introduce the relaxed notion of regularity, the so called lower-regularity, and its inheritance with respect to vertex and edge subsets.Finally, in Section 6.3.3 we state the KŁR conjecture and an analoguestatement for cycles of logarithmic length. We then use them to deriveLemma 6.14, the main lemma of this section. Later in Section 6.4we use Lemma 6.14 to deal with components of H[Q] given by thedecomposition result (Lemma 6.5).

6.3. The regularity method 99

6.3.1 Preliminaries

Let G = (V,E) be a graph and p ∈ (0, 1]. For subsets X,Y ⊆ V (notnecessarily disjoint), we define the p-density of the pair (X,Y ) as

dG,p(X,Y ) =eG(X,Y )

p|X||Y |,

and write dG(X,Y ) := dG,1(X,Y ) for the standard definition of thedensity. If the graph G is clear from the context, we omit it from thesubscript. Note that if G ∼ G(n, p), then for all sufficiently large subsetsX and Y we expect dG,p(X,Y ) = 1 ± o(1). The following definitioncaptures this observation in a more general form.

Definition 6.6 ((ε, p)-regularity). Let ε > 0 and 0 < p ≤ 1 be givenand let G = (V,E) be a graph. For disjoint subsets X,Y ⊆ V , we saythat the pair (X,Y ) is (ε, p)-regular if for all X ′ ⊆ X and Y ′ ⊆ Y with

|X ′| ≥ ε|X| and |Y ′| ≥ ε|Y |,

we have|dG,p(X,Y )− dG,p(X ′, Y ′)| ≤ ε.

If (X,Y ) is (ε, dG(X,Y ))-regular, we simply say that it is (ε)-regular.Moreover, if G is a bipartite graph whose parts form an (ε, p)-regularpair, we say that G is (ε, p)-regular.

Note that for p = 1 the definition of (ε, 1)-regularity matches the well-known definition of ε-regularity due to Szemerédi [Sze78]. Before westate a theorem which utilizes the notion of (ε, p)-regularity, we needone more definition: Given a constant ε > 0 and a graph G = (V,E),we say that a partition Viti=0 of V is (ε)-regular (for some t ∈ N) if

(i) |V0| ≤ ε|V | and |V1| = . . . = |Vt|, and

(ii) all but at most εt2 pairs (Vi, Vj) (with 1 ≤ i < j ≤ t) are (ε)-regular.

The vertex class V0 is called the exceptional set. It was observed byKohayakawa and Rödl that Szemerédi’s Regularity Lemma [Sze78] canbe adapted to the notion of (ε, p)-regularity (see, e.g., [Koh97, KR03]).Here we state a simplified version of their result due to Scott [Sco11].


Theorem 6.7 (The sparse Regularity Lemma). Given constants ε > 0and t0, ` ∈ N, there exists a constant T = T (ε, t0, `) ≥ t0 such that forany graph G = (V,E) and a partition P of V of size |P| = `, thereexists an (ε)-regular partition Viti=0 such that t0 ≤ t ≤ T0 and eachVi (for 1 ≤ i ≤ t) is contained in some set from P.

The usual choice for a partition P is to simply put P = V . However,if we know that a graph G is for example bipartite, then it is convenientto assume that every vertex class of the (ε)-regular partition is a subsetof the vertex class of G (we will use this in the proof of Lemma 6.13).For more on the sparse Regularity Lemma, see [GS05].

6.3.2 Lower regularity

It turns out that, once we are past the initial application of the sparseRegularity Lemma in the proof of Theorem 6.3, the notion of (ε, p)-regularity is too strong and consequently leads to unnecessary compli-cations. Therefore, we shall mostly work with the following one-sidedrelaxation of the (ε, p)-regularity.

Definition 6.8 ((ε, p)-lower-regularity). Let ε > 0 and 0 < p ≤ 1 begiven and let G = (V,E) be a graph. For disjoint subsets X,Y ⊆ V ,we say that the pair (X,Y ) is (ε, p)-lower-regular if for all X ′ ⊆ X andY ′ ⊆ Y with

|X ′| ≥ ε|X| and |Y ′| ≥ ε|Y |,

we have

dG,p(X′, Y ′) ≥ 1− ε.

If (X,Y ) is (ε, dG(X,Y ))-lower-regular, we say that it is (ε)-lower-regular. IfG is a bipartite graph whose parts form an (ε, p)-lower-regularpair, we say that G is (ε, p)-lower-regular.

We remark that the usual definition of (ε, p)-lower-regularity requires alower bound on the p-density of large subsets which is relative to thep-density of the pair. We could have also used this definition, howeverwe believe that the given one will make statements and calculations lesscumbersome.


Vertex subsets

The definition of (ε, p)-lower-regularity implies that if (X,Y ) is (ε, p)-lower-regular, then (X ′, Y ′) is (ε/µ, p)-lower-regular for every X ′ ⊆ X(Y ′ ⊆ Y ) of size |X ′| ≥ µ|X| (|Y ′| ≥ µ|Y |). The next proposition statesthis fact for future references.

Proposition 6.9. Let (V1, V2;E) be an (ε, p)-lower-regular bipartitegraph, for some ε > 0 and p ∈ (0, 1]. Then for any µ > ε, everypair of subsets (X1, X2) with Xi ⊆ Vi and |Xi| ≥ µ|Vi| (i = 1, 2) is(ε/µ, p)-lower-regular.

Surprisingly, Gerke, Steger, Kohayakawa and Rödl [GKRS07] showedthat if one replaces every with almost every, then the lower-regularitytypically inherits on a much finer scale then given by the definition.

Theorem 6.10 (Corollary 3.8 in [GKRS07]). Given β, ε′ ∈ (0, 1), thereexist constants ε = ε(β, ε′) > 0 and L = L(ε′) > 0 such that for anyp ∈ (0, 1] the following holds:

Let (V1, V2;E) be an (ε, p)-lower-regular bipartite graph. Then for everyq1, q2 ≥ L/p, all but at most

βminq1,q2(|V1|q1

)(|V2|q2

)pairs (Q1, Q2) with Qi ⊆ Vi and |Qi| = qi (i = 1, 2) are (ε′, p)-lower-regular.

A one-sided version of Theorem 6.10 suitable for application in randomgraphs was given in [KRSS11]. Here we state a minor modification ofthat result. It is obtained with the same proof, thus we omit it.

Lemma 6.11 (Proposition 15 in [KRSS11]). For every integer ∆ ≥ 3and constants ε′, ζ, γ, α > 0, there exist positive constants

ε∗ = ε∗(∆, ζ), ε = ε(∆, ε′, ζ) and C,

such that if p ≥ (C log n/n)1

∆−1 then G ∼ G(n, p) a.a.s satisfies thefollowing property:

For every tripartite subgraph Γ = (X,Y, Z;E) ⊆ G and q ≥ αp suchthat


(i) |X|, |Z| ≥ γnp∆−2 and |Y | ≥ γnp∆−3, and

(ii) (X,Y ) is (ε∗, q)-lower-regular and(Y,Z) is (ε, q)-lower-regular with respect to Γ,

the set of vertices with atypical neighbourhood

Γ(ε′, q) :=

x ∈ X∣∣∣∣∣∣degG(x, Y ) ≤ |Y |q/2 or(NΓ(x, Y ), Z) is not (ε′, q)-lower-regularwith respect to Γ

is of size at most ζ|X|.

Edge subsets

Some theorems, like Theorems 6.15 and 6.16 in the next section, requirean exact bound on the number of edges in a given (ε)-lower-regulargraph. The following lemma comes in handy for applying such results.

Lemma 6.12 (Lemma 4.3 in [GS05]). Given ε > 0, there exists aconstant C > 0 such that every (ε)-lower-regular bipartite graph B =(V1, V2;E) contains a (2ε)-lower-regular subgraph B′ = (V1, V2;E′) withe(B′) = m edges, for all m satisfying Cv(B) ≤ m ≤ e(B).

We remark that Lemma 4.3 in [GS05] is concerned with (ε)-regulargraphs, but one easily checks that the same proof goes through in thecase of (ε)-lower-regularity. Briefly, the idea is to show that a sub-set of m edges chosen uniformly at random satisfies lower-regularitywith positive probability. Unfortunately, most of the time we shall onlywork under assumption that a certain bipartite graph B = (V1, V2;E) is(ε, p)-lower-regular, with no control over its actual p-density. Samplinga random subset of m edges scales down the p-density of a linear sub-graph typically by m/e(B). Therefore, if e(B) is bounded away from|V1||V2|p and there exists a pair (X,Y ) with p-density very close to 1 (wecan assume both, as otherwise the graph is (ε′)-lower-regular) then the(m/|V1||V2|)-density of (X,Y ) in the sampled graph becomes boundedaway from 1 (from above) and so we lose the lower-regularity property.We overcome this by showing that every (ε, p)-lower-regular contains aspanning (ε′)-lower-regular subgraph with density close to p.

Lemma 6.13. Given ε′ > 0, there exist constants ε > 0 and C > 0 suchthat for p ∈ (0, 1], every (ε, p)-lower-regular graph B = (U1, U2;E) with


|U1| = |U2| = n and np ≥ C contains an (ε′)-lower-regular subgraphB′ = (U1, U2;E′) with e(B′) ≥ (1− ε′)n2p.

Proof. Let ξ > 0 be such that 12ξ = (ε′)3/2, and consider a partitionVii∈[t] of V (B) obtained by applying the sparse Regularity Lemmawith P = V1, V2 and ε← ξ. We prove the lemma for

ε = min

2(1− ξ)

t, ξ

and C =

C6.12(ξ)t

(1− ξ)3.

Since |V0| ≤ ξ · 2n and all sets except V0 are of the same size s := |V1|,we have (1 − ξ)2n/t ≤ s ≤ 2n/t. Consequently, the number tj of setsVi ⊆ Uj (j = 1, 2) satisfies (1− 2ξ)t/2 ≤ tj ≤ n/s. For each (ξ)-regularpair (Vi, Vj) with Vi ⊆ U1 and Vj ⊆ U2, let Eij ⊆ EB(Vi, Vj) be anedge-subset of size |Eij | = (1 − ξ)s2p such that (Vi, Vj ;Eij) is (2ξ)-lower-regular. This is indeed possible due to Lemma 6.12, since εn ≤ sand (ε, p)-lower-regularity of (U1, U2) imply

eB(Vi, Vj) ≥ (1−ε)s2p ≥ (1−ξ)s2p ≥ (1−ξ)(1−ξ)2 4n2

t2p ≥ C6.12(ξ)·2s.

The last inequality follows from the choice of C and np ≥ C. We nowdefine E′ as the union of all such Eij . Since the number of (ξ)-regularpairs (Vi, Vj) with Vi ⊆ U1 and Vj ⊆ U2 is at least t1 · t2 − ξt2 ≥(1− 8ξ)t2/4, we obtain

|E′| ≥ (1−8ξ)t2/4 ·(1−ξ)s2p ≥ (1−9ξ)t2

4·(1−ξ)2 4n2

t2p ≥ (1−11ξ)n2p,

as required (the choice of ξ gives 11ξ ≤ ε′ with room to spare). Similarcalculation gives an upper bound |E| ≤ n2p, thus the density p′ ofB′ = (U1, U2;E′) satisfies p′ ≤ p.It remains to estimate the number of edges in B′ between subsets X ⊆U1 and Y ⊆ U2 of size |X|, |Y | ≥ ε′n. Since p′ ≤ p, it suffices to show

eB′(X,Y ) ≥ (1− ε′)|X||Y |p.

For every Vi ⊆ U1 and Vj ⊆ U2 such that (Vi, Vj) is (2ξ)-regular (withrespect to B′) and |X ∩ Vi|, |Y ∩ Vj | ≥ 2ξs, we have

eB′(X ∩ Vi, Y ∩ Vj) ≥ (1− 2ξ) · |X ∩ Vi||Y ∩ Vj |(1− ξ)p.


Ignoring the regularity condition for the moment, summing over all pairs(Vi, Vj) for which |X ∩ Vi|, |Y ∩ Vj | ≥ 2ξs gives

(|X| − t1 · 2ξs− 2ξn)(|Y | − t2 · 2ξs− 2ξn)(1− 3ξ)p ≥(|X| − 4ξn)(|Y | − 4ξn)(1− 3ξ)p ≥ |X||Y |(1− 3ξ)p− 8ξn2p. (6.5)

As we might have potentially counted edges between non-regular pairs,to compensate it suffices to subtract ξt2s2p ≤ 4ξn2p from (6.5). Nowthe choice of ξ gives

12ξn2p =ε′

2(ε′)2n2p ≤ ε′

2|X||Y |p,

and with room to spare we obtain eB′(X,Y ) ≥ (1 − ε′)|X||Y |p. Thisverifies that B′ is (ε′)-lower-regular.

6.3.3 KŁR conjecture and cycles

Given a graph H, integers n,m ∈ N and a constant ε > 0, we say thata v(H)-partite graph Γ =

(⋃h∈V (H) Vh, E

)is (H,n,m, ε)-lower-regular

if

(i) |Vh| = n for every h ∈ V (Γ), and

(ii) e(Vh, Vu) = m and (Vh, Vu) is (ε)-lower-regular for every h, u ∈E(H).

We are mainly interested in finding an embedding f of H into an(H,n,m, ε)-lower-regular graph Γ such that f(h) ∈ Vh for every h ∈V (H). We call such an embedding canonical. The following lemmacomplements the decomposition result from Section 6.2 (Lemma 6.5)by providing the existence of subgraphs with certain density/structuralproperties.

Lemma 6.14. Given integers D ≥ 1 and ∆ ≥ 2 and reals α, γ > 0,there exist constants ε = ε(D,∆, α), C > 0 such that if

p ≥ (C log n/n)1

∆−1/2 ,

then G ∼ G(n, p) a.a.s has the following property: for every graph Hwith m2(H) ≤ 3/2 and


• v(H) ≤ D or

• H is a cycle of size at most 2 log n,

if Γ ⊆ G is an (H, s,m, ε)-lower-regular subgraph with

s = γnp∆−2/ log n

and m = αs2p, then there exists a canonical embedding of H into Γ.

The proof of Lemma 6.14 follows easily from Theorem 6.15 and Lemma6.16, which we state next. The following theorem was first conjecturedby Kohayakawa, Łuczak and Rödl [KŁR97]. It has attracted a lot ofattention until it was finally resolved, independently, by Saxton andThomason [ST15] and Balogh, Morris and Samotij [BMS15] (see also[CGSS14] for a slightly different statement).

Theorem 6.15 (KŁR conjecture). Given a graph H and a constantβ > 0, there exist constants ε0, C > 0 and n0 ∈ N such that for n ≥ n0,m ≥ Cn2−1/m2(H) and ε ∈ (0, ε0], for all but at most

βm(n2

m

)e(H)

(H,n,m, ε)-lower-regular graphs Γ there exists a canonical embedding ofH into Γ.

We remark that the theorem proven in [BMS15, ST15] deals with thecase where the pairs of Γ corresponding to edges of H are (ε)-regular,rather than (ε)-lower-regular. However, careful analysis of their proofsreveals that only the number of edges m and lower-regular propertiesare used, thus it implies the version stated here.

One of the drawbacks of Theorem 6.15 is that it can only be applied tofixed graphs. To deal with cycles of logarithmic size in Lemma 6.14 weneed the following statement about expansion in lower-regular graphs,due to Gerke et al. [GKRS07]. To state it concisely, we use the followingnotion of expansion: given ` ∈ N and δ > 0, we say that a graphG = (V,E) is (δ, `)-expanding for subsets A,B ⊆ V if for at least(1 − δ)|A| vertices a ∈ A there exist at least (1 − δ)|B| vertices inB which are at distance at most ` from a. In the next lemma, we useP` to denote the path on ` vertices.


Lemma 6.16 (Lemma 5.9, [GKRS07]). Given integer ` ≥ 2 and con-stants β, δ > 0, there exist constants ε0, C > 0 and n0 ∈ N such that forall n ≥ n0, m ≥ Cn2−(`−2)/(`−1) and ε ∈ (0, ε0], all but at most

βm(n2

m

)`−1

(P`, n,m, ε)-lower-regular graphs Γ = (V1, . . . , V`;E) are

(δ, `− 1)-expanding

for V1 and V`.

With Theorem 6.15 and Lemma 6.16 at hand, we are ready to proveLemma 6.14.

Proof of Lemma 6.14. Let β = (2α/e2)D2

and assume that ε ∈ (0, 1/2)is smaller than any ε0(H,β) given by Theorem 6.15 with v(H) ≤ D andε0(` = 4, β, δ = 2/3) given by Lemma 6.16.

We first show that G ∼ G(n, p) a.a.s does not contain any of the ex-cluded lower-regular configurations from Theorem 6.15. Let H be agraph on at most D vertices and m2(H) ≤ 3/2. A simple calcu-lation shows that for sufficiently large enough C > 0 we have p ≥C6.15(H,β)s−2/3/α and consequently

m ≥ αs2p ≥ C6.15(H,β)s2−m2(H).

Therefore, the expected number of copies of (H, s,m, ε)-lower-regulargraphs excluded by Theorem 6.15 is at most(

n

s

)v(H)

βm(s2

m

)e(H)

pe(H)m ≤ ev(H)s lognβm(es2p

m

)e(H)m

< (eβ)m( eα

)e(H)m

= o(1).

In the last inequality we used a simple estimate e(H) ≤ D2. Markov’sinequality implies that a.a.s none of these copies appear in G. Moreover,since there are only constantly many graphs on at most D vertices, itfollows from the union-bound that the conclusion of the lemma holdsfor every graph H with v(H) ≤ D and m2(H) ≤ 3/2.


Without loss of generality we may assume D ≥ 7. It remains to provethe case when H = Ct is a cycle of size t ∈ [7, 2 log n]. Similar calcula-tion as in the previous case shows that every (P4, s,m, ε)-lower-regularsubgraph (V1, . . . , V4;E) ⊆ G is (2/3, 3)-expanding for V1 and V4. Con-sider an (Ct, s,m, ε)-lower-regular subgraph Γ = (V1, . . . , Vt;E) ⊆ G.From the previous conclusion we know that there exists a vertex v ∈ V4

and subsets X1 ⊆ V1 and X7 ⊆ V7 of size at least 2s/3 such that thereis a path of length 3 between every x ∈ X1 ∪X7 and v. Therefore, forevery x ∈ X1 and x′ ∈ X7 there exists a path of length 6 in Γ using onevertex from each X2, . . . , X6. We now define the set Xi (i = 8, . . . , t) asXi := NΓ(Xi−1, Vi). From (ε)-lower-regularity of (Vi−1, Vi) and the sizeof X7 we infer that each set Xi is of size at least 2s/3. Moreover, theconstruction implies that for every x ∈ X1 and x′ ∈ Xt there exists acanonical path in Γ\E(V1, Vt) from x to x′. Finally, from the (ε)-lower-regularity of (Vt, V1) we conclude that there exists an edge between Xt

and Xi, which closes a cycle. This finishes the proof of the lemma.

6.4 Proof of the main theorem

The proof of Theorem 6.3 utilizes the regularity method and closelyfollows the proof given in [KRSS11]. We have already stated the mainidea in Section 6.3, but we repeat it here in more details for the conve-nience of the reader. We start with arbitrary colouring of the edges ofG ∼ G(N, p) with red and blue. This naturally splits G into a red andblue subgraph denoted by Gr and Gb, respectively. Applying the sparseRegularity Lemma on Gr, we obtain a vertex partition Vii∈[t] suchthat all but at most εt2 pairs (Vi, Vj) are (ε)-regular (with respect toGr). From the tight concentration of the edges in G(N, p), we concludethat all such pairs are also (ε)-regular in Gb. At this point we con-sider an auxiliary graph on the vertex set [t] and an edge ij if (Vi, Vj)is (ε)-regular (in either of the colours). This gives an almost completegraph and by Turán’s theorem we obtain a subset S′ ⊆ [t] such thatevery two vertices in S′ correspond to an (ε)-regular pair in both Grand Gb. Next, for each i, j ∈ S′, we assign a colour to the edge ijwhich corresponds to the majority colour of the edges between Vi andVj . From Ramsey’s theorem we then obtain a colour (say red) and asubset S′ ⊆ S of size |S′| = ∆2 + 1, such that (Vi, Vj) is (ε)-regular inGr for each i, j ∈ S′. Moreover, the density of each such pair in Gr isroughly half of the density in G (this follows from taking the majority


colour). Finally, the following lemma shows that these properties implythat Gr[S′] is F∆(n)-universal, which finishes the proof.

Lemma 6.17 (The Embedding Lemma). Given integer ∆ ≥ 5 andconstants α, µ > 0, there exist constants ε = ε(∆), C, β > 0 such thatif p ≥ (C log n/n)

1∆−1/2 then G ∼ G(n, p) a.a.s has the property that

every (K∆2+1, s,m, ε)-lower-regular subgraph Γ ⊆ G with s ≥ µn andm ≥ αs2p is F∆(βn)-universal.

Note that the only difference between our proof and the one in [KRSS11]is in Lemma 6.17. The proof of an analogue statement in [KRSS11]transfers the idea of Alon and Füredi [AF92] to the sparse regularitysetting. However, this strategy effectively embeds a graph vertex byvertex, which enforces a lower bound of order (log n/n)1/∆ (see Section6.1.1 for a detailed discussion). Instead, our proof is inspired by theideas from Chapter 5. It combines the decomposition result from Section6.2 (Lemma 6.5), corollary of the KŁR conjecture from Section 6.3.3(Lemma 6.14) and the embedding scheme and analysis from Section3.2.

Before we prove Lemma 6.17, we first use it to derive Theorem 6.1.

Proof of Theorem 6.3. We start by defining necessary constants. LetR = r(K∆2+1) be the smallest integer with the property that ev-ery 2-colouring of the edges of KR contains a monochromatic copy ofK∆2+1. Set ξ = ε6.17(∆), ε = ε6.13(ξ)/2, T = T6.7(t0 = R, ε) andβ = β6.17(∆, α = 1/4, µ = (1− ε)/T ). Moreover, we can assume that εis such that 2ε < 1/(R−1)−1/R. We prove the theorem for B = d1/βeand sufficiently large C > 0 such that G = (V,E) ∼ G(N, p) a.a.ssatisfies the following properties:

• the property of Proposition 2.4 with µ← ε(1− ε)/T and

• the property of the Embedding Lemma (Lemma 6.17) with µ ←(1 − ε)/T and α ← 1/4 (all parameters which are not specifiedtake values as defined above).

Let E1 ∪ E2 be arbitrary edge-colouring of G and let Gi = (V,Ei)(i = 1, 2) be the subgraph defined by colour i. Consider an (ε)-regularpartition Viti=0 (for R ≤ t ≤ T ) obtained by applying the sparseRegularity Lemma (Theorem 6.7) with G1. Since |V0| ≤ εN , each


Vi (i ≥ 1) is of size |Vi| = s ≥ (1−ε)NT . Note that if (Vi, Vj) is (ε)-

regular in G1 then by Proposition 2.4 it is also (2ε)-regular in G2. Thesparse Regularity Lemma guarantees that the number of (ε)-regularpairs (Vi, Vj) in G1 is at least(

t

2

)− εt2 ≥ (1− 1

t− 2ε)

t2

2> (1− 1

R− 1)t2

2.

The last inequality follows from the assumption 2ε < 1/(R − 1)− 1/Rand t ≥ R. Therefore, by Turan’s theorem [Tur41] there exists a subsetI ⊆ 1, . . . , t of size |I| ≥ R such that (Vi, Vj) is (ε)-regular in G1 forevery i, j ∈ I. Next, consider a complete graph K|I| on the vertex set Iand for each i, j ∈ I colour the edge ij with the colour 1 if eG1(Vi, Vj) ≥eG2(Vi, Vj) and with the colour 2 otherwise. From the size of I we inferthat there exists c ∈ 1, 2 and a subset I ′ ⊆ I of size |I ′| = ∆2 + 1such that every edge ij with i, j ∈ I ′ is of the colour c. In other words,for every i, j ∈ I ′ we have the following:

• (Vi, Vj) is (2ε)-regular in Gc and

• eGc(Vi, Vj) ≥ eG(Vi, Vj)/2 ≥ (1− ε)s2p/2 ≥ s2p/3.

The second property follows from the choice of the colour c for theedge ij and the Proposition 2.4. Therefore, (Vi, Vj) is (2ε, p/3)-lower-regular in Gc. Furthermore, from Lemma 6.13 we have that for eachi, j ∈ I ′ there exists Eij ⊆ EGc(Vi, Vj) of size |Eij | = s2p/4 such that(Vi, Vj ;Eij) is (ξ)-lower-regular (we implicitly assume that ξ is suffi-ciently small such that (1− ξ)1/3 > 1/4). Therefore, the subgraph

Γ =

⋃i∈I′

Vi,⋃i,j∈I′

Eij

⊆ Gis (K∆2+1, s, s

2p/4, ξ)-lower-regular and by the Embedding Lemma it isF∆(βN)-universal. Since all edges in Γ are coloured with c, this finishesthe proof of the theorem.

Finally, we prove Lemma 6.17.

Proof of Lemma 6.17. We start by defining constants and properties ofG used in the proof. Let k ∈ N be the smallest integer such that

k − 2− ∆− 2

∆− 1/2(k − 1) > 0, (6.6)


and set

ζ =1

4k2∆2, γ = µ(α/2)∆/(4k), ε2 = ε6.14(D = 5∆2,∆, α/2),

ε1 = ε6.13(ε2), ε∗ = minε∗6.11(∆, ζ), 1/(16∆), ε6.10(β = 1/2, ε1),ξ2∆ = ε∗

ξd = minξd+1, ε6.11(∆, ε′ = ξd+1, ζ) for d ∈ 2∆− 1, . . . , 0.

We prove the lemma for ε = ξ0/k and β = (γ/8)2. The choice ofconstants will become apparent in the proof. Moreover, we assume thatG satisfies the following properties:

(P1) the property of the Proposition 2.5 with λ ← γ/8 and all d ∈1, . . . ,∆− 2,

(P2) the property of the Lemma 6.11 with ε ← ξd and ε′ ← ξd+1 ford ∈ 0, . . . , 2∆− 1,

(P3) the property of the Lemma 6.14 with D ← 5∆2 and α← α/2.

All parameters which we did not explicitly specify in (P1)–(P3) are usedas defined above. Note that for sufficiently large C > 0 and p as statedin the lemma, G ∼ G(n, p) a.a.s satisfies these properties.

Let Γ = (V1, . . . , V∆2+1;E) ⊆ G be an (K∆2+1, s,m, ε)-lower-regulargraph for some s ≥ µn and m ≥ αs2p. Note that for q = αp every pairof subsets (Vi, Vi′) is (ε, q)-lower-regular in Γ. For each i ∈ [∆2 + 1]choose arbitrary (nearly) equitable-partition Vi = V 1

i ∪ . . .∪ V ki and lets′ denote the size of V ji . Since all inequalities in the proof will hold witha large margin, we can safely assume s′ = µn/k. Then, by Proposition6.9, for every j, j′ ∈ [k] and i 6= i′ the pair (V ji , V

j′

i′ ) is (ξ0, q)-lower-regular.

Next, consider a graph H ∈ F∆(βn) and let ϕ : V (H) → [∆2 + 1] be aproper colouring of H2. We aim to find an embedding ψ : V (H)→ V (Γ)of H into Γ such that ψ(v) ∈ Vϕ(v) for every v ∈ V (H). Consider apartition

V (H) =

⋃i∈[t]

(ri ∪ Si)

∪Qgiven by Lemma 6.5 and let Q1, . . . , Qt′ denote vertex sets of compo-nents in H[Q]. We embed H into Γ in t+ t′ steps: in step z ∈ 1, . . . , t


we embed vertices rz ∪ Sz and in step z′ ∈ t+ 1, . . . , t+ t′ verticesfrom Qz′−t. Let Dz ⊆ V (H) (for z ∈ 0, . . . , t + t′) denotes the setof vertices embedded after the first z steps (with D0 = ∅) and let Ezdenotes the set of edges among vertices which are not yet embedded, i.e.Ez := E(H[V (H) \Dz]). Importantly, properties (a)–(c) of the Lemma6.5 imply

|NH(v,Dz)| ≤ ∆− 2 for every z ∈ 0, . . . , t+ t′ and v ∈ V (H) \Dz.(6.7)

We now describe the embedding procedure.

Let ψz : Dz → V (Γ) be the embedding of H[Dz] into Γ obtained afterthe first z steps. For each vertex v ∈ V (H) \Dz and j ∈ [k] we definethe subset of candidates Ajz(v) ⊆ V jϕ(v) for a vertex v as follows,

Ajz(v) = V jϕ(v) ∩ NΓ(ψz(NH(v,Dz)))

= V jϕ(v) ∩⋂

w∈NH(v,Dz)

NΓ(ψz(w), V jϕ(v)).

In particular, for each vertex v ∈ V (H) \ Dz and j ∈ [k], if Ajz(v) \ψz(Dz) 6= ∅ then any extension of ψz to v with ψz(v) ∈ Ajz(v)\ψz(Dz)is an embedding ofH[Dz∪v] into Γ. The main challenge is to maintainsets Ajz(v) such that whenever we are about to embed a vertex v we canfind some j ∈ [k] for which Ajz(v) \ ψz(Dz) 6= ∅. We show that byalways choosing the smallest such j ∈ [k] this will indeed be the case.Observant reader will notice that this is exactly the same strategy weused in Section 3.2.

As already said, we proceed in t + t′ steps. We maintain the followingtwo properties after each step z ∈ 0, . . . , t:

(i) for each v ∈ V (H) \Dz and j ∈ [k] we have

|Ajz(v)| ≥ s′(q/2)|NH(v,Dz)| ≥ 4γnp|NH(v,Dz)|

(ii) for each vw ∈ Ez and each j, j′ ∈ [k], the pair (Ajz(v), Aj′

z (w)) is(ξd, q)-lower-regular where d = |NH(v,Dz)|+ |NH(w,Dz)|.

Since Aj0(v) = V jϕ(v) and each pair (V ji , Vj′

i′ ) is (ξ0, q)-lower-regular,properties (i) and (ii) hold for z = 0. We now describe in details stepsz = 1, . . . , t:


1. For each v ∈ rz∪Sz let jv ∈ [k] be the smallest index such that|Ajvz−1(v)\ψz−1(Dz−1)| ≥ |Ajvz−1(v)|/2 (we postpone the proof thatsuch jv exists).

2. For each v ∈ Sz, let Bv ⊆ Ajvz−1(v) (the bad vertices) denotes theset of vertices such that having ψz(v) ∈ Bv implies that either (i)or (ii) will be violated for some vertex or an edge:

Bv =⋃

x,y∈[k]

⋃vw∈Ezw 6=rz

⋃wu∈Ezu6=v

Γxyvwu(ξduw+1, q),

where Γxyvwu ⊆ Γ denotes the tripartite graph induced by Ajvz−1(v),Axz−1(w) and Ayz−1(u), duw = |NH(u,Dz−1)|+ |NH(w,Dz−1)| andΓxyvwu(ξduw+1, q) is as defined in Lemma 6.11. We show that

|Bv| < |Ajvz−1(v)|/4.

Let x, y ∈ [k] and consider a vertex v ∈ Sz and edges vw,wu ∈ Ezwith w 6= rz and u 6= v. From the property (c) of Lemma 6.5(the decomposition lemma) we conclude w ∈ Q. Moreover, sinceδ(H[Q]) ≥ 2 and (Q∪v)∩Dz−1 = ∅, we have |NH(w,Dz−1)| ≤∆− 3. Together with (6.7) and the assumption that (i) holds forz−1 this implies |Ajvz−1(v)|, |Ayz−1(u)| ≥ γnp∆−2 and |Axz−1(w)| ≥γnp∆−3. On the other hand, from the property (ii) for z − 1 wehave that (Ajvz−1(v), Axz−1(w)) is (ε∗, q)-lower-regular (since ε∗ ≥ξd for every d ∈ [2∆]) and (Axz−1(w), Ayz−1(u)) is (ξduw , q)-lower-regular. Therefore, we can apply the property (P2) to conclude|Γxyvwu(ξduw+1, q)| ≤ ζ|Ajvz−1(v)|. Going over all x, y ∈ [k] andthe edges vw,wu ∈ Ez with w 6= rz and u 6= vs, this implies|Bv| ≤ k2∆2ζ|Ajvz−1(v)| < |Ajvz−1(v)|/4.

Consequently, the subset of vertices Lv = Ajvz−1(v)\(ψz−1(Dz−1)∪Bv) has the property that having ψz(v) ∈ Lv gives an embeddingof H[Dz−1∪v] and (i) and (ii) continue to hold for every vertexand edge influenced by v (i.e. every vertex or an edge touchingNH(v)\ (Dz−1∪rz)). Moreover, from the bound on Bv and thechoice of jv in 1. we get

|Lv| ≥ |Ajvz−1(v)|/4. (6.8)


3. Let v = rz. Using the same notation as in the previous step, wedefine Bv, B′v ⊆ A

jvz−1(v) correspondingly,

Bv =⋃

x,y∈[k]

⋃vw∈Ezw 6=Sz

⋃wu∈Ezu6=v

Γxyvwu(ξduw+1, q),

B′v =⋃w∈Sz

a ∈ Ajvz−1(v) : NΓ(a, Lw) = ∅.

The role of Bv is the same as in 2. and B′v guarantees that havingψz(rz) /∈ B′v allows w ∈ Sz to be embedded into Lw. We showthat

|Bv ∪B′v| < |Ajvz−1(v)|/2.

By following the same argument as in 2., one easily obtains |Bv| <|Ajvz−1|/4. The only difference is in the proof of |NH(w,Dz−1)| ≤∆−3, but since this can be done by a simple case analysis (eitherw ∈ Q, w ∈ Sz′ or w = rz′ for some z′ > z) we omit it. On theother hand, from the property (ii) for z−1, the bound in (6.8) andthe Proposition 6.9 we have that (Ajvz−1(v), Lw) is (4ε∗, q)-lower-regular for each w ∈ Sz. It is an easy exercise to show that thedefinition of (4ε∗, q)-lower-regularity then implies

|B′v| ≤ |Sz| · 4ε∗|Ajvz−1(v)| < |Ajrzz−1(rz)|/4.

The subset of vertices Lv = Ajvz−1(v) \ (ψz−1(Dz−1) ∪ B′v ∪ Bv)then has the property that ψz(v) ∈ Lv allows each w ∈ Sz tobe embedded into its subset of good vertices Lw, and (i) and (ii)continue to hold for vertices and edges influenced by v (the sameas in 2.). The choice of jv in 1. implies Lv 6= ∅.

4. Finally, we define ψz to be arbitrary extension of ψz−1 to rz∪Szwith ψz(rz) ∈ Lrz and ψz(v) ∈ NH(ψz(rz), Lv) for each v ∈ Sz.The property (d) of the Lemma 6.5 (the decomposition result)implies that for each edge uw ∈ Ez and j ∈ [k] either Ajz(u) =Ajz−1(u) or Ajz(w) = Ajz−1(w) (or both) and for each w ∈ V (H) \Dz there exists at most one edge vw ∈ E(H) with v ∈ rz ∪Sz. Moreover, ψz is indeed an embedding since ϕ(v) 6= ϕ(v′) fordistinct v, v′ ∈ rz ∪ Sz (follows from the fact that ϕ is a propercolouring of H2), and consequently Lv ∩ Lv′ = ∅. Using theseobservations, it is straightforward to check that the definition ofLv for v ∈ rz ∪ Sz implies properties (i) and (ii) for z.


In steps z ∈ t + 1, . . . , t + t′ we no longer have to worry aboutmaintaining properties (i) and (ii). Reason for this is that for everyz ∈ t+1, . . . , t+t′ and v ∈ V (H)\Dz we have NH(v,Dz) = NH(v,Dt)(follows from a trivial fact that there are no edges between different con-nected components) and therefore,

Ajz(v) = Ajt (v)(6.7)≥ 4γnp∆−2

for every j ∈ [k]. Using this observation and ξd ≤ ε∗ for d ≤ 2∆, for theremainder of the proof we only use that (Ajz(v), Aj

′

z (w)) is (ε∗, q)-lower-regular for every j, j′ ∈ [k] and vw ∈ Ez. We now repeat the followingfor z = t+ 1, . . . , t+ t′:

1. For each v ∈ Qz−t let jv ∈ [k] be the smallest index such thatAjv (v) := Ajvz−1(v) \ ψz−1(Dz−1) is of size at least |Ajv (v)| ≥|Ajvz−1(v)|/2 (again, we defer the proof that such jv exists);

2. For each v ∈ Qz−t let A(v) ⊆ Ajv (v) be a subset of size sa =γnp∆−2/ log n such that the following holds:

• A(v) ∩A(w) = ∅ for distinct v, w ∈ Qz−t and

• if vw ∈ H[Qz−t] then the pair (A(v), A(w)) is (ε1, q)-lower-regular.

The existence of such sets can be seen as follows: First, place eachvertex a ∈

⋃v∈Qz−t Ajv (v) into a set A(v) uniformly at random

among all v with a ∈ Ajv (v). Since each vertex a ∈ Ajv (v) we putin A(v) with probability at least 1/(2 log n) (the maximal size ofa component in H[Q]), by the Chernoff’s inequality and a unionbound we get |A(v)| ≥ |Ajv (v)|/(4 log n) ≥ sa with high proba-bility, for each v ∈ Qz−t. By further random sampling we makeeach A(v) to be of size exactly sa. Note that the whole proce-dure w.h.p chooses uniformly a subset of Avj (v) of size sa. Since(Ajv (v), Ajw(w)) is (ε∗, q)-lower-regular for each vw ∈ H[Qz−t]and ϕ(v) 6= ϕ(w) implies A(v) and A(w) are chosen mutuallyindependently, from sa 1/q and Theorem 6.10 we have that(A(v), A(w)) is (ε1, q)-lower-regular with high probability. More-over, this probability is sufficiently high to do a union-bound overall edges in H[Qz−t]. This shows the existence of subsets A(v).


3. For each vw ∈ H[Qz−t], by Lemma 6.12 there exists an (ε2)-lower-regular pair (A(v), A(w);Evw) with |Evw| = αs2

ap/2. Observe thatthe graph

Γ′ =

⋃v∈Qz−t

A(v),⋃

vw∈H[Qz−t]

Evw

is then (H[Qz−t], sa, αs

2ap/2, ε2)-lower-regular.

4. From the property (P3) applied with Γ← Γ′, there exists a canon-ical copy of H[Qz−t] into Γ′. Finally, we define ψz to be an ex-tension of ψz−1 to Qz−t given by such copy. It is straightforwardto verify that ψz is an embedding of H[Dz−1 ∪Qz−t] into Γ.

Since Dt+t′ = V (H) we have that ψ := ψt+t′ is an embedding of Hinto Γ. It remains to prove that 1. is well-defined in both cases (z ≤ tand z > t). We show that for every j ∈ 1, . . . , k, i ∈ [∆2 + 1] andz ∈ 0, . . . , t+ t′ we have

|ψz(Dz) ∩ V ji | ≤ g(j), where g(j) =

βn, if j = 1g(j−1)4∆3

γnp∆−2 , if j ≥ 2.(6.9)

Assuming that this is the case, from

|Akz(v)| ≥ γnp∆−2 ≥ 1 and |ψz(Dz) ∩ V kϕ(v)| ≤ g(k)(6.6)< 1

for every z ∈ 0, . . . , t + t′ and v ∈ V (H) \Dz, we conclude that thechoice of jv in 1. is always possible. We now show that (6.9) alwaysholds.

Let us assume, towards the contradiction, that (6.9) is false for some j ∈1, . . . , k, i ∈ [∆2 + 1] and z ∈ 0, . . . , t+ t′. Moreover, assume that jis the smallest index for which this is the case. Since v(H) ≤ βn = g(1),we know j ≥ 2. Let J ⊆ Dz be the subset of all vertices v ∈ Dz withψz(v) ∈ V ji and, for each v ∈ J , use zv ∈ 1, . . . , z to denote the stepbefore the one in which the vertex v was embedded. Observe that foreach v ∈ J we have NH(j,Dzv ) 6= ∅, as otherwise A1

zv (v) = V 1ϕ(v) and

|V 1ϕ(v) \ Dz| ≥ |V 1

ϕ(v)|/2 imply v is embedded into V 1ψ(v) (contradicting

j ≥ 2). Therefore, we infer from ∆(H) ≤ ∆ and (6.7) that there existsd ∈ 1, . . . ,∆− 2 and a subset J ′ ⊆ J of size at least

|J ′| ≥ |J |/∆3 > g(j)/∆3 ≥ g(j − 1)4

γnpd


such that the following holds:

• |NH(v,Dzv )| = d for every v ∈ J ′ and

• NH(v,Dzv ) ∩NH(w,Dzw) = ∅ for distinct v, w ∈ J ′.

The existence of such J ′ can be shown, for example, by first choosingd ∈ [∆−2] for which at least |J |/∆ vertices v ∈ J have |NH(v,Dzv )| = dand then greedily picking those which do not overlap. Moreover, we canassume that J ′ is exactly of size g(j − 1)4/(γnpd). Owing to the choiceof β, we have |J ′| < γ/(8pd). The crucial observation is that ψz(v) ∈ V jiand the definition of jv in 1. imply

|Aj−1zv (v) ∩ ψz(Dz)| ≥ |Aj−1

zv (v) ∩ ψzv (Dzv )| ≥ |Aj−1zv (v)|/2

(i)> γnpd,

(6.10)for every v ∈ J ′, as otherwise v would be embedded into V j−1

i . WritingMv := ψzv (Nh(v,Dzv )) for v ∈ J ′, from

Aj−1zv (v) ⊆ NΓ(Mv) ⊆ NG(Mv) (6.11)

and the property (P1) we further have∣∣∣NG(Mv) \ (Aj−1zv (v) ∩ ψz(Dz))

∣∣∣ (6.10)≤

∣∣∣NG(Mv)∣∣∣− γnpd

(P1)

≤ (1− γ/2)npd. (6.12)

Applying again the property (P1) to estimate the size of⋃v∈J′ NG(Mv),

we finally obtain

|ψz(Dz) ∩ V j−1i | ≥

∣∣∣∣∣ ⋃v∈J′

Aj−1zv (v) ∩ ψz(Dz)

∣∣∣∣∣(6.11)

=

∣∣∣∣∣ ⋃v∈J′

NG(Mv) \(NG(Mv) \ (Aj−1

zv (v) ∩ ψz(Dz))∣∣∣∣∣

≥

∣∣∣∣∣ ⋃v∈J′

NG(Mv)

∣∣∣∣∣−∣∣∣∣∣ ⋃v∈J′

NG(Mv) \ (V j−1i ∩ ψz(Dz))

∣∣∣∣∣(P1),(6.12)

> (1− γ/8)|J ′|npd − |J ′|(1− γ/2)npd

> |J ′|γnpd/4 = g(j − 1),


which is a contradiction with the minimality of j. To conclude, weshowed that (6.9) always holds which, by the earlier discussion, impliesthat 1. is well-defined. This finishes the proof of the lemma.


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