Chromatic Ramsey Number and Circular Chromatic Ramsey Number

1. Chromatic Ramsey Number and Circular Chromatic Ramsey Number Xuding Zhu Department of Mathematics Zhejiang Normal University

2. Among 6 people, There are 3 know each other, or 3 do not know each other. Know each other Do not know each other

3. Among 6 people, There are 3 know each other, or 3 do not know each other.

4. Among 6 people, There are 3 know each other, or 3 do not know each other.

5. Among 6 people, There are 3 know each other, or 3 do not know each other.

6. 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.

7. 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:

8. A sufficiently largescale (or complicated) systemmust contains an interesting sub-system. “Complete disorder is impossible”

9. 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.

10. 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” .

11. 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.

12. Ramsey number R(3,k) Furstenberg [124] gave ergodic theoretical and topological dynamics reformulations.

13. means. For any 2-colouring of the edges of F with coloursredandblue, there is a red copy ofGor a blue copy ofH.

14. The Ramsey number of (G,H) is

15. 1933, George Szekeres, Esther Klein, Paul Erdos starting with a geometric problem, Szekeres re-discovered Ramsey theorem, and proved

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

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

19. Bounds for R(k,l)

20. Bounds for R(k,l)

21. Bounds for R(k,l)

22. 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?

24. Chromatic number Circular chromatic number

25. G=(V,E): a graph 0 an integer 1 1 An k-colouring of G is 2 0 such that A 3-colouring of

26. The chromatic number of G is

27. 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

28. The circular chromatic number of G is { r: G has a circular r-colouring } min inf

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

30. 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

31. and Basic relation between Circular chromatic number of a graph is a refinement of its chromatic number.

32. Graph coloring is a model for resource distribution Circular graph coloring is a model for resource distribution of periodic nature.

33. Introduced by Burr-Erdos-Lovasz in 1976

34. 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.

35. 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

36. 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)

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

38. Graph homomorphism = edge preserving map H G

39. 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.

40. 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.

41. H G GxH Projections are homomorphisms

42. 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.

43. H G

45. 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.

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

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

48. Fractional Hedetniemi’s conjecture

49. 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.

50. 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