Improved algorithms for colorings of simple hypergraphs and applications

Dmitry Shabanovjoint work with

Jakub Kozik

Workshop on Extremal Graph Theory, June 06, Yandex

Definitions• A hypergraph is a vertex set and a family of subsets whose

elements are called the edges of the hypergraph.• A hypergraph is said to be -uniform if every edge consists of

exactly vertices.• Let be a hypergraph. A vertex coloring is called proper for if

there is no monochromatic edges in this coloring.• A hypergraph is said to be -colorable if there is a proper

coloring with colors for it.• The chromatic number of the hypergraph , denoted by , is the

minimum number of colors required for a proper coloring.

Colorings of Graphs

The bound is tight (e.g., complete graphs or odd cycles).

This bound is tight up to the constant factor.

Basic fact

Any graph with maximum vertex degree has chromatic number at most .

Theorem (A. Johansson, 1996)

Any triangle-free graph with maximum vertex degree has chromatic number at most .

What about hypergraphs?• Let be a hypergraph. The degree of an edge is the number of

other edges of intersecting . The maximum edge degree of is denoted by .

This theorem was historically the first application of the famous Local Lemma.

Theorem (P. Erdős, L. Lovász, 1973)

If is a -uniform hypergraph with maximum edge degree at most

then is -colorable (i.e. .

Theorem for 2-colorings

The proof is based on the random recoloring method.

Theorem (J. Radhakrishnan, A. Srinivasan, 2000)

If is an -uniform hypergraph with maximum edge degree at most

then is -colorable.

Recent progress

The proof is based on Pluhár’s criterion for -colorability of an arbitrary hypergraph in terms of ordered -chains.

UPDATE: Theorem holds for arbitrary number of colors with replaced by an absolute constant

Theorem (D. Cherkashin, J. Kozik, 2013)

Suppose is fixed. Then for any sufficiently large , if is an -uniform hypergraph with at most

then is -colorable.

Colorings of simple hypergraphs• A hypergraph is called simple if every two of its distinct edges

have at most one common vertex (as in graphs), i.e. for any .It turns out that it is somehow easier to color simple hypergraphs.

Theorem (Z. Szabó, 1990)

For any , there is such that for any , the following statement holds: if is an -uniform simple hypergraph with at most

then is -colorable.

Recent improvementsTheorem (A. Kupavskii, D. Shabanov, 2013)

For any , if is an -uniform simple hypergraph with

then is -colorable.

Theorem (J. Kozik, 2013)

If and is an -uniform simple hypergraph with

then is -colorable.

Main new resultThe proofs of the mentioned theorems appeared to be “orthogonal”. The joined efforts led to the following result.

Theorem holds for any and as in the classical Theorem of Erds and Lovász.

Theorem (J. Kozik, D. Shabanov, 2014+)

If is an -uniform simple hypergraph with maximum edge degree at most

where is an absolute constant, then is -colorable.

Comparison with other results

• We proved that any -uniform simple non--colorable hypergraph satisfies

• If is a constant then this lower bound is times smaller than

the upper bound given by A. Kostochka and V. Rödl who showed that there exists an -uniform simple non--colorable hypergraph with

• For large , A. Frieze and D. Mubayi established that any -uniform simple non--colorable hypergraph satisfies

with

Property B conjectureLet denote the minimum possible number of edges in an -uniform non--colorable hypergraph.

Similar problem: Let denote the minimum possible value of the maximum edge degree in an -uniform non--colorable hypergraph.Conjecture: We proved the lower bound in the class of simple hypergraphs.

Conjecture (P. Erdős, L. Lovász, 1973)

𝑚 (𝑛)=Θ (𝑛2𝑛) .

Ingredients of the proof

• Random recoloring method

• Almost complete analysis of the recoloring procedure (h-tree construction)

• Special variant of the Local Lemma

Random recoloring methodSuppose is an -uniform simple hypergraph. Without loss of generality assume that .

Let be independent random variables with uniform distribution on , weights of the vertices. Let be a real number.

A vertex is called a free vertex if . Only free vertices are allowed to recolor during the recoloring procedure.

FIRST STAGE Color every vertex randomly and independently with colors. The obtained coloring is called initial.

Recoloring procedureSECOND STAGE 1. Start with initial coloring.

2. If in the current coloring there is a monochromatic edge (of some color ) containing a free vertex which has not been recolored yet, then– take a free vertex of with initial color and the least

weight ;– recolor with color .

3. Repeat step 2 until there is no monochromatic edges with

non-recolored free vertices.

Recoloring procedure

In such situation we say that the third vertex blames the edge .

Construction of an h-treeSuppose that recoloring procedure fails and in the final coloring there is a monochromatic edge (root of the directed tree) of some color . Then every vertex of • either has initial color and is not free• or has initial color , is free and was recolored with during the

recoloring process

Every vertex of the second type blames some other edge (choose one for every vertex). Let be these edges. We add them to the h-tree as neighbors of .

Construction of an h-treeEvery edge from became completely monochromatic of a color at some step of the recoloring procedure. So, in the initial coloring can contain the vertices of a color which were recolored with . Every such vertex blames some edges (choose one for every vertex), we add them as neighbors of in the h-tree. Continue the process if possible.OBSERVATIONS1. The edges are different for every .2. The edges and can coincide for different . 3. The leaves of the h-tree aremonochromatic in the initial coloring.

Analysis of bad eventsThere could be the following configurations in the h-tree.

1. The edges of the h-tree form a real hypertree.

2. There are cycles in the h-tree. In this case we take the smallest subtree containing a cycle. Then either there is a short cycle (of length) or there is a large acyclic subtree (of length).

Local LemmaTheorem (Local Lemma, polynomial style)

Suppose that are independent random variables and are events from the algebra generated by them. Let denote the smallest set of variables such that . Denote for ,

Suppose that there exists a polynomial such that for every and . If, moreover, there is a real number such that , then .

Application: Van der Waerden Number

The function from the Van der Waerden Theorem is called the Van der Waerden function or the Van der Waerden number.

Question: how can we estimate ?

Theorem (B. Van der Waerden, 1927)

For any integers , there exists the smallest number such that in any -coloring of the set of integers there is a monochromatic arithmetic progression of length .

Known bounds for W(n,r) The best general upper bound was obtained by W.T. Gowers (2001):

In the particular cases the best results are due to T. Sanders (2011), B. Green and T. Tao (2009).

Known lower bounds are very far away from Gowers’ tower.

E. Berlekamp (1968) , is a prime.

Z. Szabó (1990) , provided

New lower bound for W(n,r)

This improves the previous results for of the type .

When the number of colors is large in comparison with progression length (say, ), a better lower bound can be obtained by using Hypergraph Symmetry Theorem.

Theorem (J. Kozik, D. Shabanov, 2014+)

For any ,

where is an absolute constant.

Ideas of the proof• We have to show that the hypergraph of arithmetic

progressions (vertex set , edges are arithmetic progressions of length ) is -colorable.

• This hypergraph of arithmetic progressions is not simple, but (in some sense) close to be simple.

• Its codegree is at most (not 1, as in simple hypergraphs) which is sufficient for our probabilistic construction.

• Use the same random recoloring procedure.• We have to deal with 2-cycles in h-trees, especially with situations when there are two edges with a lot of common vertices ().