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Chromatic Ramsey Number and ircular Chromatic Ramsey Numb Xuding Zhu Department of Mathematics Zhejiang Normal University

Chromatic Ramsey Number and Circular Chromatic Ramsey Number

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Chromatic Ramsey Number and Circular Chromatic Ramsey Number. Xuding Zhu. Department of Mathematics Zhejiang Normal University. Among 6 people,. There are 3 know each other, or 3 do not know each other. Know each other. Do not know each other. Among 6 people,. - PowerPoint PPT Presentation

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Page 1: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Chromatic Ramsey Number andCircular Chromatic Ramsey Number

Xuding Zhu

Department of MathematicsZhejiang Normal University

Page 2: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Among 6 people,

There are 3 know each other, or 3 do not know each other.

Know each other

Do not know each other

Page 3: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Among 6 people,

There are 3 know each other, or 3 do not know each other.

Page 4: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Among 6 people,

There are 3 know each other, or 3 do not know each other.

Page 5: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Among 6 people,

There are 3 know each other, or 3 do not know each other.

Page 6: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Among 6 people,

There are 3 know each other, or 3 do not know each other.

Colour the edges of by red or blue,there is either a red or a blue

6K3K 3K

Page 7: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Theorem [Ramsey] For any graphs G and H, there exists a graph F such that if the edges of F are coloured by red and blue, then there is a red copy of G or a blue copy of H

For `any’ systems , there exists a system F such that if `elements’ of F are partitioned into k parts, then for some i, the ith part contains as a subsystem.

k)i (1 i G

iG

General Ramsey Type Theorem:

Sufficiently large or complicated

Page 8: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

“Complete disorder is impossible”

A sufficiently large scale (or complicated) system must contains an interesting sub-system.

Page 9: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

There are Ramsey type theorems in many branches of mathematics such as combinatorics, number theory, geometry, ergodic theory, topology, combinatorial geometry, set theory, and measure theory.

Ramsey Theory has a wide range of applications.

Page 10: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

If the k-tuples M are t-colored, then

Theorem [Ramsey, 1927]

,,, knt,|| with set aFor mMM

nMMM |'| ,'

all the k-tuples of M’ having the same color.

principle pigeonhole :1k

graph complete a of edges thecolouring :2k

m

Page 11: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

For any partition of integers into finitely many parts, one part contains arithematical progression of arbitrarylarge length.

Van der Waeden Theorem

Szemerédi's theorem (1975)

Every set of integers A with positive density contains

arithematical progression of arbitrary length.

Timonthy Gowers [2001] gave a proof using both Fourier analysis and combinatorics.

Regularity lemmaErdos and Turan conjecture (1936)

Harmonic analysis

Page 12: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Ramsey number R(3,k)

Page 13: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

For any 2-colouring of the edges of F with colours red and blue,there is a red copy of G or a blue copy of H.

),(K 336 KK

),( HGF means.

),(K 335 KK

Page 14: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

),( :min HGKnR(G,H) n

),(),( lkRKKR lk

The Ramsey number of (G,H) is

),( :min ,3 3 kn KKKnk)R(

Page 15: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

1933, George Szekeres, Esther Klein, Paul Erdos

starting with a geometric problem, Szekeres re-discovered Ramsey theorem, and proved 2 ,3 kk)R(

Page 16: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Erdos [1946]

3/2

lnk

k ,3 k)R(

Erdos [1961]

Graver-Yackel [1968]

k

kOk)R(

ln

lnln k ,3 2

2

lnk

k ,3 k)R(

Ajtai-Komlos-Szemeredi[1980]

k

kOk)R(

ln ,3

2

Kim [1995]

k

kk)R(

ln ,3

2

2 ,3 kk)R( Szekere [1933]

Many sophisticated probabilistic tools are developed

Page 17: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

George Szekere and Esther Klein marriedlived together for 70 year, died on the same day 2005.8.28, within one hour.

Page 18: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

),( :min lkn KKKnR(k,l)

Page 19: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Bounds for R(k,l)

k l 3 4 5 6 7 8

3 6 9 14 18 23 28

4 18 25 36

41

49

61

58

84

5 43

49

58

87

80

143

101

216

6 102

165

113

298

132

495

7 205

540

217

1031

8 282

1870

Page 20: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Bounds for R(k,l)

k l 3 4 5 6 7 8

3 6 9 14 18 23 28

4 18 25 36

41

49

61

58

84

5 43

49

58

87

80

143

101

216

6 102

165

113

298

132

495

7 205

540

217

1031

8 282

1870

Page 21: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Bounds for R(k,l)

k l 3 4 5 6 7 8

3 6 9 14 18 23 28

4 18 25 36

41

49

61

58

84

5 43

49

58

87

80

143

101

216

6 102

165

113

298

132

495

7 205

540

217

1031

8 282

1870

Page 22: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

How to measure a system?

A sufficiently large scale (or complicated) system must contains an interesting sub-system.

What is large scale?

What is complicated?

How to measure a graph?

Page 23: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

),( : min

is ,( ofnumber Ramsey The

HGKnR(G,H)

H)G

n

),( |:)(| min

is ,( ofnumber Ramsey The

HGFFVR(G,H)

H)G

),( |:)(| min

is ,( ofnumber Ramsey Size The

HGFFE(G,H)R

H)G

E

),( |:)(| min

is ,( ofnumber Ramsey -degree-max The

HGFF(G,H)R

H)G

),( :)( min

is ,( ofnumber Ramsey chromatic The

HGFF(G,H)R

H)G

),( :)( inf

is ,( ofnumber Ramsey chromaticcircular The

HGFF(G,H)R

H)G

cc

Page 24: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Chromatic number

Circular chromatic number

Page 25: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

1,...,1,0: kVf

)()(~ yfxfyx

G=(V,E): a graph

an integer:1k

3k

An k-colouring of G is

0

1

20

1

A 3-colouring of 5Csuch that

Page 26: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

The chromatic number of G is

colouring-k a has G :kmin )( G

Page 27: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Vf :

)()( yfxf

G=(V,E): a graph

an integer:1k

k-colouring of G is

such that yx ~

An

1|)()(|1 kyfxf

a real number

A (circular)

1,...,1,0 k),0[ k

0

1

20.5

1.5

A 2.5-coloring

1r

r-colouring of G is

),0[ r

1|)()(|1 ryfxf

Page 28: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

The circular chromatic number of G is

)(Gc { r: G has a circular r-colouring }infmin

Page 29: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

f is k-colouring of G

Therefore for any graph G,

)()( GGc

f is a circular k-colouring of G

Page 30: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

0=r

3

1

24

x~y |f(x)-f(y)|_r ≥ 1

The distance between p, p’ in the circle is

|'| |,'| min|p'-p| r pprpp

f is a circular r-colouring if

0 r

pp’

Page 31: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Basic relation between )(G and )(Gc

).()(1)( GGG c

Circular chromatic number of a graph is a refinementof its chromatic number.

Page 32: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Graph coloring is a model for resource distribution

Circular graph coloring is a model for resource distributionof periodic nature.

Page 33: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

),( :)( min

is ,( ofnumber Ramsey chromatic The

HGFF(G,H)R

H)G

Introduced by Burr-Erdos-Lovasz in 1976

),( lkR),K(KR lk

),( , HGRH)(GR

),( GGR(G)R

Page 34: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

If F has chromatic number , then there is a2 edge colouring of F in which each monochromaticsubgraph has chromatic number n-1.

2)1( n

),( GGF for any n-chromatic G.

4n

Page 35: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

If F has chromatic number , then there is a2 edge colouring of F in which each monochromaticsubgraph has chromatic number n-1.

2)1( n

),( GGF for any n-chromatic G.

1)1()( then ,)( If :nObservatio 2 nGRnG

1)1()( and )(G with graph a is there

n,each For :1976] Lovasz,-Erdos-[Burr Conjecture2 nGRnG

Could be much larger

Page 36: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

1)1()( and )(G with graph a is there

n,each For :1976] Lovasz,-Erdos-[Burr Conjecture2 nGRnG

The conjecture is true for n=3,4 (Burr-Erdos-Lovasz, 1976)

The conjecture is true for n=5 (Zhu, 1992)

The conjecture is true (Zhu, 2011)

Attempts by Tardif, West, etc. on non-diagonal casesof chromatic Ramsey numbers of graphs.

There are some upper bounds on nGG )(:)(Rmin

No more other case of the conjecture were verified, until 2011

1)1()(:)(Rmin 2 nnGG

Page 37: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

)hom(Kn

Lovasz]-Erdos-[Burr Lemma

n G(G)R

For any 2 edge-colouring of Kn, there is a monochromaticgraph which is a homomorphic image of G.

Page 38: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Graph homomorphism = edge preserving map

GH

Page 39: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

1)1()( and )(G with graph a is there

n,each For :1976] Lovasz,-Erdos-[Burr Conjecture2 nGRnG

To prove Burr-Erdos-Lovasz conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.

1)1( 2nK

The construction of G is easy:

Take all 2 edge colourings of 1)1( 2nK

mccc ,,, 21

For each 2 edge colouring ci of , one of the monochromaticsubgraph, say Gi, , has chromatic number at least n.

1)1( 2nK

iG

Page 40: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.

1)1( 2nK

The construction of G is easy:

Take all 2 edge colourings of 1)1( 2nK

mccc ,,, 21

For each 2 edge colouring of , one of the monochromaticsubgraph, say Gi, , has chromatic number at least n.

1)1( 2nK

iGic

mGGGG 21

Page 41: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

HG

G

H

GxH

Projections are homomorphisms

Page 42: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.

1)1( 2nK

The construction of G is easy:

Take all 2 edge colourings of 1)1( 2nK

mccc ,,, 21

For each 2 edge colouring ci of , one of the monochromaticsubgraph, say Gi, , has chromatic number at least n.

1)1( 2nK

mGGGG 21

GGi of image chomomorphi a is Each

iG

?

Page 43: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

G

H

Page 44: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

)(),(min)(

:1966] i,[Hedetniem Conjecture

HGHG

Page 45: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.

1)1( 2nK

?

If Hedetniemi’s conjecture is true, then

Burr-Erdos-Lovasz conjecture is true.

Page 46: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

GG of setst independen offamily :)(

A k-colouring of G partition V(G) into k independent sets.

1,0)(: G

1)()(,

GXXv

X

)(

)(minGX

X)(G

integer linear programming

Page 47: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

GG of setst independen offamily :)(

A k-colouring of G partition V(G) into k independent sets.

1,0)(: G

1)()(,

GXXv

X

)(

)(minGX

X)(G

]1,0[

)(Gf

ofnumber chromatic fractional :)( GGf

linear programming

)()( GGf

Page 48: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

)(),(min)(

:1966] i,[Hedetniem Conjecture

HGHG

)(),(min)(

:2002] [Z, Conjecture

HGHG fff

Fractional Hedetniemi’s conjecture

Page 49: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.

1)1( 2nK

If Hedetniemi’s conjecture is true, then

Burr-Erdos-Lovasz conjecture is true.

Observation: If fractional Hedetniemi’s conjecture is true, then Burr-Erdos-Lovasz conjecture is true.

Page 50: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.

1)1( 2nK

The construction of G is easy:

Take all 2 edge colourings of 1)1( 2nK

mccc ,,, 21

For each 2 edge colouring ci of , one of the monochromaticsubgraph, say Gi, , has chromatic number at least n.

1)1( 2nK

mGGGG 21

GGi of image chomomorphi a is Each

iG fractional chromatic number > n-1

1)( nGf 1)()( nGG f

Page 51: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

)(),(min)(

:1966] i,[Hedetniem Conjecture

HGHG

)(),(min)(

:2002] [Z, Conjecture

HGHG fff

Fractional Hedetniemi’s conjecture

)(),(min)(

:2002] [Z, Conjecture

HGHG fff

Theorem [Huajun Zhang, 2011]

If both G and H are vertex transitive, then

Theorem [Z, 2011]

Page 52: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

GG of setst independen offamily :)(

A k-colouring of G partition V(G) into k independent sets.

1,0)(: G

1)()(,

GXXv

X

)(

)(minGX

X)(G

]1,0[

)(Gf

ofnumber chromatic fractional :)( GGf

linear programming

]1,0[)(: GV

dual problem

1)()(,

GXXv

v

)(

)(maxGVv

v)(Gf

ofnumber clique fractional :)( GGf

Page 53: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

The fractional chromatic number of G is obtained by solving a linear programming problem

The fractional clique number of G is obtained by solving its dual problem

)()( GG ff

Page 54: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

)(),(min)(

:2002] [Z, Conjecture

HGHG fff

Fractional Hedetniemi’s conjecture is true

Theorem [Z, 2010]

)(),(min)( HGHG fff

:sketch oofPr

)(),(min weight with total

of clique fractional aconstruct tosuffices

HG

HG

ff

)(),(min)( HGHG fff

)(),(min)( HGHG fff

)(),(min)( HGHG fff

Easy!

Difficult!

Page 55: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

)(),(min weight with total

of clique fractional aconstruct tosuffices

HG

HG

ff

Easy

Easy

Difficult

Page 56: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

G of clique fractional maximum a ,10 : ],[V(G)g

H of clique fractional maximum a ],1,0[)( : HVh

(H)(G),ωω

g(x)h(y)(x,y)

HGV

ffmax

as defined ],1,0[)( :

)(),(min weight with total

of clique fractional a is

HG

HG

ff

Uyxff HGyhxg

HGU

),(

)(),(max)()(

, of set t independen

Easy!

Difficult!

Page 57: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

),( :)( inf

is ,( ofnumber Ramsey chromaticcircular The

HGFF(G,H)R

H)G

cc

G)(GR(G)Rcc

,

What is the relation between and ? (G)Rc c(G)

Page 58: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

Basic relation between )(G and )(Gc

).()(1)( GGG c

).()( GGc

)(Gc )(Gis a refinement of

)(G )(Gcis an approximation of

Page 59: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

There are many periodical scheduling problems in computer sciences.

The reciprocal of is studied by computer scientists as efficiency of a certain scheduling method, in 1986.

).(Gc

Circular colouring is a good model for periodical scheduling problems

Page 60: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

zGGRzR ccc )(:)(inf)(

nGGRnR )(:)(min)(

1)1()( :1976] Lovasz,-Erdos-[Burr Conjecture 2 nnR Theorem [Zhu, 2011]

?)( zRc No conjecture yet!

Page 61: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

).()()( GGG cf

.1)()( GG c

.largey arbitraril becan )()( GG f

)1( )( then ,integeran is 2 If kkkRkc

Using fractional version of Hedetniemi’s conjecture,Jao-Tardif-West-Zhu proved in 2014

)12(2 )2( c

R )13(3 )( 3 KRc

?)13(3 )3( c

R

Page 62: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

2 rational real becan )( Gc

?)( of valuespossible theareWhat x

GR

Page 63: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

),( :)( inf GGFF(G)R cc

colouring-circular a has : inf rGr(G)c min

min ?

No !

[ Jao-Tardif-West-Zhu, 2014]

Page 64: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

4 )( then ,2

52 If zRz

c

6 )( 3 KRc

9 ),( 43 KKRc

5 ),(3

14 53 CCR

c

2

9 ),(4 73 CCR

c

Some other results by Jao-Tardif-West-Zhu, 2014

Page 65: Chromatic Ramsey Number and Circular Chromatic Ramsey Number

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