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Chromatic Ramsey Number and Circular Chromatic Ramsey Number

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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,

There are 3 know each other,

or 3 do not know each other.

Among 6 people,

There are 3 know each other,

or 3 do not know each other.

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

Among 6 people,

There are 3 know each other,

or 3 do not know each other.

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.

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

Sufficiently large or complicated

General Ramsey Type Theorem:

A sufficiently largescale (or complicated) systemmust contains

an interesting sub-system.

“Complete disorder is impossible”

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.

Theorem [Ramsey, 1927]

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

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

whenever

the

elements

of

some

(suﬃciently

large)

object

are

partitioned

into

a

ﬁnite

number

of

classes

(i.e.,

colored

with

a

ﬁnite

number

of

colors),

there

is

always

at

least

one

(color)

class

which

contains

all

the

elements

of

some

regular

structure.

When

this

is

the

case,

one

additionally

would

like

to

have

quantitative

estimates

of

what

“suﬃciently

large”

means.

In

this

sense,

the

guiding

philosophy

of

Ramsey

theory

can

be

described

by

the

phrase:

“Complete

disorder

is

impossible”

.

Van der Waeden Theorem

For any partition of integers into finitely many parts,

one part contains arithematical progression of arbitrary

large length.

Regularity lemma

Erdos and Turan conjecture (1936)

Szemerédi's theorem (1975)

Every set of integers A with positive density contains

arithematical progressionof arbitrary length.

Harmonic analysis

Timonthy Gowers[2001] gave a proof

using both Fourier analysis and combinatorics.

Ramsey number R(3,k)

Furstenberg

[124]

gave

ergodic

theoretical

and

topological

dynamics

reformulations.

means.

For any 2-colouring of the edges of F with coloursredandblue,

there is a red copy ofGor a blue copy ofH.

The Ramsey number of (G,H) is

1933, George Szekeres, Esther Klein, Paul Erdos

starting with a geometric problem, Szekeres re-discovered

Ramsey theorem, and proved

Szekere [1933]

Erdos [1946]

Erdos [1961]

Graver-Yackel [1968]

Ajtai-Komlos-Szemeredi

[1980]

Kim [1995]

Many sophisticated probabilistic tools are developed

George Szekere and Esther Klein married

lived together for 70 year,

died on the same day 2005.8.28, within one hour.

A sufficiently largescale (or complicated) systemmust contains

an interesting sub-system.

How to measure a system?

What is large scale?

What is complicated?

How to measure a graph?

Chromatic number

Circular chromatic number

G=(V,E): a graph

0

an integer

1

1

An k-colouring of G is

2

0

such that

A 3-colouring of

The chromatic number of G is

G=(V,E): a graph

0

a real number

1

an integer

1.5

A (circular)

k-colouring of G is

r-colouring of G is

An

2

0.5

A 2.5-coloring

such that

The circular chromatic number of G is

{ r: G has a circular r-colouring }

min

inf

f is k-colouring of G

f is a circular k-colouring of G

Therefore for any graph G,

0=r

0

r

1

4

2

3

|f(x)-f(y)|_r ≥ 1

x~y

p

p’

The distance between p, p’ in the circle is

f is a circular r-colouring if

and

Basic relation between

Circular chromatic number of a graph is a refinement

of its chromatic number.

Graph coloring is a model for resource distribution

Circular graph coloring is a model for resource distribution

of periodic nature.

Introduced by Burr-Erdos-Lovasz in 1976

If F has chromatic number , then there is a

2 edge colouring of F in which each monochromatic

subgraph has chromatic number n-1.

for any n-chromatic G.

If F has chromatic number , then there is a

2 edge colouring of F in which each monochromatic

subgraph has chromatic number n-1.

for any n-chromatic G.

Could be much larger

There are some upper bounds on

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

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

Attempts by Tardif, West, etc. on non-diagonal cases

of chromatic Ramsey numbers of graphs.

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

The conjecture is true (Zhu, 2011)

For any 2 edge-colouring of Kn, there is a monochromatic

graph which is a homomorphic image of G.

Graph homomorphism = edge preserving map

H

G

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.

The construction of G is easy:

Take all 2 edge colourings of

For each 2 edge colouring ci of , one of the monochromatic

subgraph, say Gi, , has chromatic number at least n.

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.

The construction of G is easy:

Take all 2 edge colourings of

For each 2 edge colouring of , one of the monochromatic

subgraph, say Gi, , has chromatic number at least n.

H

G

GxH

Projections are homomorphisms

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.

?

The construction of G is easy:

Take all 2 edge colourings of

For each 2 edge colouring ci of , one of the monochromatic

subgraph, say Gi, , has chromatic number at least n.

H

G

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.

?

If Hedetniemi’s conjecture is true, then

Burr-Erdos-Lovasz conjecture is true.

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

integer linear programming

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

linear programming

Fractional Hedetniemi’s conjecture

Observation: If fractional Hedetniemi’s conjecture is true,

then

Burr-Erdos-Lovasz conjecture is true.

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.

If Hedetniemi’s conjecture is true, then

Burr-Erdos-Lovasz conjecture is true.

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.

The construction of G is easy:

Take all 2 edge colourings of

For each 2 edge colouring ci of , one of the monochromatic

subgraph, say Gi, , has chromatic number at least n.

fractional chromatic number > n-1

Theorem [Huajun Zhang, 2011]

If both G and H are vertex transitive, then

Fractional Hedetniemi’s conjecture

Theorem [Z, 2011]

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

dual problem

linear programming

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

Fractional Hedetniemi’s conjecture is true

Theorem [Z, 2010]

Easy!

Difficult!

Easy

Easy

Difficult

Difficult!

Easy!

What is the relation between and ?

is a refinement of

and

Basic relation between

is an approximation of

The reciprocal of is studied by computer

scientists as efficiency of a certain

scheduling method, in 1986.

Circular colouring is a good model for periodical

scheduling problems

There are many periodical scheduling problems in

computer sciences.

Theorem [Zhu, 2011]

No conjecture yet!

Using fractional version of Hedetniemi’s conjecture,

Jao-Tardif-West-Zhu proved in 2014

min

?

min

No !

[ Jao-Tardif-West-Zhu, 2014]

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

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