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.

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

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. General Ramsey Type Theorem: Sufficiently large or complicated

“Complete disorder is impossible” A sufficiently large scale (or complicated) system must contains an interesting sub-system.

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.

whenevertheelementsofsome (su ffi ciently large)objectare partitionedintoafinitenumberofclasses(i.e.,coloredwithafinitenumber ofcolors),thereisalwaysatleastone(color)classwhichcontainsallthe elementsofsomeregularstructure.Whenthisisthecase,oneadditionally wouldliketohavequantitativeestimatesofwhat “su ffi ciently large”means. Inthissense,theguidingphilosophyofRamseytheorycanbedescribedby thephrase:“Completedisorderisimpossible”. If the k-tuples M are t-colored, then Theorem [Ramsey, 1927] all the k-tuples of M’ having the same color.

For any partition of integers into finitely many parts, one part contains arithematical progression of arbitrary large 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 lemma Erdos and Turan conjecture (1936) Harmonic analysis

Furstenberg[124]gaveergodictheoreticalandtopological dynamics reformulations. Ramsey number R(3,k)

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. means.

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

Erdos [1946] Erdos [1961] Graver-Yackel [1968] Ajtai-Komlos-Szemeredi [1980] Kim [1995] Szekere [1933] Many sophisticated probabilistic tools are developed

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

Bounds for R(k,l) k l

Bounds for R(k,l) k l

Bounds for R(k,l) k l

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?

Chromatic number Circular chromatic number

G=(V,E): a graph an integer An k-colouring of G is A 3-colouring of such that

The chromatic number of G is

G=(V,E): a graph an integer k-colouring of G is such that An a real number A (circular) A 2.5-coloring r-colouring of G is

The circular chromatic number of G is { r: G has a circular r-colouring } infmin

f is k-colouring of G Therefore for any graph G, f is a circular k-colouring of G

0=r x~y |f(x)-f(y)|_r ≥ 1 The distance between p, p’ in the circle is f is a circular r-colouring if 0r p p’p’

Basic relation between and 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

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 cases of chromatic Ramsey numbers of graphs. There are some upper bounds on No more other case of the conjecture were verified, until 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 G H

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.

G H 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. ?

G H

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

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. 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. 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

Fractional Hedetniemi’s conjecture Theorem [Huajun Zhang, 2011] If both G and H are vertex transitive, then Theorem [Z, 2011]

A k-colouring of G partition V(G) into k independent sets. linear programming dual problem

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 Difficult

Easy! Difficult!

What is the relation between and ?

Basic relation between and is a refinement of is an approximation of

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 Circular colouring is a good model for periodical scheduling problems

Theorem [Zhu, 2011] No conjecture yet!

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

min ? No ! [ Jao-Tardif-West-Zhu, 2014]

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

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