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

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 ` > 3 people? Are these numbers finite?

3. Ramsey Theory Extensions: Color graphs other than cliques (e.g., random graphs). Avoid some fixed graph F other than K `. Avoid graph F 1 in blue and F 2 in red (asymmetric case). Allow more colors. Ramsey (1930)

4. Random Graphs Binomial model G n, p n vertices include each edge with probability p, independently of all other edges We study the limiting probability that the random graph G n, p satisfies a given property P, where p = p ( n ). It turns out that many properties have threshold functions p 0 ( n ) such that

5. Ramsey properties R ( F, k ) 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. 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).

6. Threshold for Ramsey properties Intuition: above the threshold, there are more copies of F in G n, 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 (!) Łuczak, Ruciński, Voigt (1992) / Rödl, Ruciński (1993, 1995)

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

8. Threshold for asymmetric Ramsey properties 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. Conjecture: Kohayakawa, Kreuter (1997)

9. Threshold for asymmetric Ramsey properties (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. Conjecture: Kohayakawa, Kreuter (1997) The KŁR-Conjecture is known to be true for trees, cycles, and cliques of size at most 5.

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

11. About the proof

12. Overview Our proof is along similar lines as the original proof for the symmetric case by Rödl and Ruciński. Need to show: with high probability, in each coloring of G n, p there is either a blue copy of H or a red copy of G. We show by induction on e ( G ): with very high probability, in each coloring of […] (e.g. G n, p ) there is either a blue copy of H or many […] red copies of G. denote this event by R We solve one key issue in a fundamentally different way than in the original proof. also yields a simpler proof for the symmetric case

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

14. 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. The issue many (by induction) not too many !?

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 D holds, 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): Maybe deleting a few edges suffices to ensure that there are not too many copies of D in the remaining graph. denote this event by D * Deletion Lemma: Pr[ : D *] is indeed „very small“ Robustness Lemma: Deleting a few edges does not mess up the rest of the proof. i.e. we can still show We get

17. Alternative solution Use the FKG inequality! A graph property is called increasing if it cannot be destroyed by adding edges containing a C 4 is an increasing property containing an induced C 4 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 G n, 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, :A increasing.

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 D are decreasing events!

21. 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. Marciniszyn, Skokan, S., Steger (2006) / Kohayakawa, Schacht, S. (2009)

22. Thank you! Questions?