Upper bounds for asymmetric Ramsey properties of random graphs

1. Upper bounds for asymmetric Ramsey properties of random graphs Reto Spöhel, ETH Zürich Joint work with Yoshiharu Kohayakawa, Universidade de São PauloMathias Schacht, Universität Hamburg TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

2. Ramsey Theory • Folklore • Among every party of at least six people, there are at least three, either all or none of whom know each other • Equivalently:Every edge-coloring of the complete graph on six vertices with two colors contains a monochromatic triangle. • Question: • How many people must attend the party so that the assertion holds for ` > 3people? • Are these numbers finite?

3. Ramsey Theory Ramsey (1930) • Extensions: • Color graphs other than cliques (e.g., random graphs). • Avoid some fixed graph Fother thanK`. • Avoid graph F1 in blue and F2 in red (asymmetric case). • Allow more colors.

4. Random Graphs • Binomial model Gn,p • n vertices • include each edge with probability p, independently of all other edges • We study the limiting probabilitythat the random graph Gn,p satisfies a given property P, where p=p(n). • It turns out that many properties have threshold functionsp0(n) such that

5. Ramsey properties R(F, k) • Problem: For any fixed graph F, integer k, and edge probability p=p(n), determine • Observation: • The family of graphs satisfying R(F, k) is increasing. •  The property R(F, k) has a threshold (Bollobás, Thomason, 1987). By R(F, k) we denote the family of all graphs that contain a monochromatic copy of F in every edge coloring with k colors.

6. Łuczak, Ruciński, Voigt (1992)/Rödl, Ruciński (1993, 1995) Threshold for Ramsey properties • Intuition: • above the threshold, there are more copies of Fin Gn,p than edges. • This forces the copies of F to overlap substantially and makes coloring difficult. • Order of magnitude of threshold does not depend on k (!)

7. R(G, H) ByR(G, H) we denote the family of all graphs that contain either a red copy of Gor a blue copy of Hin every edge coloringwith red and blue. Asymmetric Ramsey properties • What happens is if we want to avoid differentgraphs Fi in different colors i, 1·i·k? • We focus on the case with two colors.

8. Threshold for asymmetric Ramsey properties Conjecture: Kohayakawa, Kreuter (1997) Kohayakawa, Kreuter (1997) The conjecture is true if G and H are cycles. Marciniszyn, Skokan, S., Steger (RANDOM’06) The 0-statement is true if G and H are cliques.

9. Threshold for asymmetric Ramsey properties Conjecture: Kohayakawa, Kreuter (1997) (e.g.) Marciniszyn, Skokan, S., Steger (RANDOM’06) The 1-statement is true if H satisfies some balancedness condition and the KŁR-Conjecture holds for G. • The KŁR-Conjecture is known to be true for trees, cycles, and cliques of size at most 5.

10. Threshold for asymmetric Ramsey properties Conjecture: Kohayakawa, Kreuter (1997) Kohayakawa, Schacht, S. (2009+) The 1-statement is true if H satisfies some balancedness condition and the KŁR-Conjecture holds for G. • …. in particular for the case where G and H are cliques of arbitrary size.

11. About the proof

12. Overview • Ourproofisalongsimilarlines as the original proofforthesymmetriccaseby Rödl and Ruciński. • Need toshow: withhighprobability, in eachcoloringofGn,pthereiseither a bluecopyofHor a redcopyofG. • Weshowbyinduction on e(G): withveryhighprobability, in eachcoloringof[…] (e.g. Gn,p) thereiseither a bluecopyof H ormany […] redcopiesofG. • denotethiseventbyR • Wesolveonekeyissuein a fundamentally different waythan in the original proof. • also yields a simpler proofforthesymmetriccase

13. „Small“ and „very small“ • Terminology: • „small“: something like • „very small“: something like

14. The issue • Goal: show that Pr[:R] is very small (needed for induction). • At some point in the proof, we need that there are not too many copies of some graph D. many (byinduction) not toomany !?

15. The issue • Goal: show that Pr[:R] is very small (needed for induction). • At some point in the proof, we need that there are not too many copies of some graph D. • denote this event by D • Assuming that Dholds, we can show that the probability for :R is indeed „very small“. • i.e. we can show • Issue: Pr[:D] is „small“, but not „very small“! • We only get

16. The deletion method • Rödl and Ruciński (1995): Maybedeleting a fewedgessufficestoensurethattherearenot toomanycopiesofD in theremaining graph. • denotethiseventbyD* • Deletion Lemma: Pr[:D*] isindeed „verysmall“ • Robustness Lemma:Deleting a fewedgesdoes not mess uptherestoftheproof. • i.e. wecan still show • Weget

17. Alternative solution • Use the FKG inequality! • A graph property is called increasing if it cannot be destroyed by adding edges • containing a C4 is an increasing property • containing an inducedC4 is not an increasing property.

18. Alternative solution • Use the FKG inequality! • A graph property is called increasing if it cannot be destroyed by adding edges • not true for arbitrary distributions! • in particular not for Gn,m ! Fortuin, Kasteleyn, Ginibre (1971)

19. Alternative solution • Similarly: • A graph property is called decreasing if it cannot be destroyed by removing edges • A decreasing ,:Aincreasing. Fortuin, Kasteleyn, Ginibre (1971)

20. Alternative solution • Goal: show that Pr[:R] is „very small“ • Can show: Pr[:R Æ D] is „very small“ • Issue: Pr[:D] is „small“, but not „very small“! • Observation: Both :R and Dare decreasing events!

21. Marciniszyn, Skokan, S., Steger (2006) / Kohayakawa, Schacht, S. (2009) Summary • We proved an upper bound on the threshold for a large class of asymmetric Ramsey properties. • Together with earlier results, we obtain in particular • In our proof we replaced the deletion method by an „FKG shortcut“. This also yields a simpler proof for the original symmetric case. • Our proof seems to extend to more than two colors, but this needs considerable extra work.

22. Thank you!Questions?