1. Asymmetric Ramsey Properties of Random Graphs involving Cliques Reto Spöhel Joint work with Martin Marciniszyn, Jozef Skokan, and Angelika Steger 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 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 thresholds functions p 0 ( n ) such that

5. Ramsey properties R ( F, k ) Denote the family of all graphs that contain a monochromatic copy of graph F in every edge coloring with k colors by 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 monotone 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. Threshold for Ramsey properties Łuczak, Ruciński, Voigt (1992) / Rödl, Ruciński (1993, 1995) It is conjectured that the constants b and B can be replaced by b 0 (1 §  ) for some b · b 0 · B (“sharp threshold”). verified for trees (Friedgut, Krivelevich, 2000) and F = K 3, k = 2 (Friedgut et al., 2006, 100+ pages!) [sort of: it remains open whether b 0 = b 0 ( n ) is a constant!]

8. Asymmetric Ramsey properties R ( L, R ) Denote the family of all graphs that contain either a red copy of graph L or a blue copy of graph R in every edge coloring with red and blue by R ( L, R ). 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.

9. Threshold for asymmetric Ramsey properties Kohayakawa, Kreuter (1997) The conjecture is true if L and R are cycles. Marciniszyn, Skokan, S., Steger (RANDOM’06) The 0-statement is true if L and R are cliques. The 1-statement is true if L and R are cliques (and KŁR-conjecture holds). Conjecture: Kohayakawa, Kreuter (1997)

10. Proving the 0-Statement

11. The 0-Statement 0-Statement Our proof is constructive: We propose an algorithm that computes a valid coloring of G n, p a.a.s. Our algorithm also covers the symmetric case (with some exceptions) and the asymmetric case involving cycles. All previous proofs were non-constructive.

12. The coloring algorithm Algorithm proceeds in 2 phases:  remove edges from G one by one in some clever way  insert the edges back in the reverse order, always maintaining a valid coloring Basic Idea: edges which are not exactly the intersection of an ` -clique with an r -clique can be colored directly when reinserted in Phase 2.  successively remove such edges in Phase 1. Example: coloring without a red K 4 nor a blue K 5

13. The coloring algorithm Advanced Idea: In Phase 2, we don’t need to color the inserted edge directly, but may recolor an existing edge first.  We can ignore cliques which contain an edge that can be recolored. Example: coloring without a red K 4 nor a blue K 5 (ctd.) In fact, the recoloring of an existing edge (from red to blue) can be postponed to later in Phase 2 by pushing the ` -clique on a stack in Phase 1. This includes the original idea: the edge we recolor later (from red to blue) may be the edge we just inserted.

14. The coloring algorithm Phase 2: Work through S and if S.top is an edge: insert and color red if S.top is a copy of K ` : check if all red, recolor one edge to blue if needed. Phase 1: Remove edges successively and build stack S containing copies of K ` we don’t care about anymore (because they contain an edge which can be recolored), and edges which are not in a copy of K ` we still care about At the end of Phase 1, all edges and all copies of K ` are on the stack. ………

15. The coloring algorithm If all edges are removed in Phase 1, Phase 2 produces a valid coloring: No red copy of K ` is created, because every copy of K ` is examined in phase 2, and one edge is recolored to blue if needed. No blue copy of K r is created, because the recoloring never creates a blue copy of K r. The algorithm can be shown to remove all edges from G n, p in Phase 1 a.a.s. unless ` = 3.  Algorithm needs to be refined for triangles. Phase 2: Work through S and if S.top is an edge: insert and color red if S.top is a copy of K ` : check if all red, recolor one edge to blue if needed. Phase 1: Remove edges successively and build stack S containing copies of K ` we don’t care about anymore, and edges which are not in a copy of K ` we still care about

16. The trouble with triangles If p  bn -1/ m 2 ( K 3, K r ), then G = G n, p a.a.s. contains K r + 1. Algorithm gets stuck in Phase 1 since every edge of K r + 1 is the intersection of K 3 and K r (and there is no copy of K 3 with an edge that can be recolored). Even nastier substructures may appear. Solution: Determine those structures. Color them separately at the beginning of phase 2. Example: ` = 3, r = 4

17. Proof sketch To do: Show that a.a.s all edges will be removed in Phase 1 ( ` > 3 ) or only easily colorable graphs remain ( ` = 3). Proof idea: Graphs for which algorithm gets stuck contain substructures that do not appear in G n, p. If algorithm gets stuck on some graph G, every edge of G is contained in a copy of K `, every edge of which is contained in an otherwise disjoint copy of K r, every edge of which...  Use these properties to build a subgraph of G which is either too dense or too large to appear in G n, p Lemma For p  bn -1/ m 2 ( K `, K r ), the Coloring Algorithm terminates a.a.s. and produces a valid coloring of G n, p.

18. Proof Sketch Sunflowers: copies of K `, each edge of which is intersection with a copy of K r. Example: r = 5 and ` = 4 Every copy of K ` that is not the center of a sunflower contains an edge that can be recolored.  If algorithm gets stuck on G, every copy of K ` is center of a sunflower. However: outer K r ’s may mutually overlap! Deterministic Lemma: If algorithm fails in Phase 1, then G contains either a dense structure of sunflowers (much overlap), or a large structure of sunflowers (little overlap). Probabilistic Lemma: Both events are very unlikely to happen in G n, p.

19. Remarks The generalization to asymmetric Ramsey properties with more than two colors is straightforward, as the conjectured threshold depends only on the two densest graphs F i. It follows from known results that the proposed algorithm also works for the symmetric case (with some exceptions) and the asymmetric case involving cycles. What about the general asymmetric case? It seems plausible that the proposed algorithm works in most cases. Two main challenges: deal with overlapping sunflowers determine graphs which may remain after Phase 1

20. About the 1-Statement

21. The 1-Statement 1-Statement The proof is routinely obtained using Szemerédi’s Regularity Lemma for sparse graphs (Kohayakawa, Rödl, 1997), and the KŁR-Conjecture (1997), a probabilistic embedding lemma for ( , p ) -regular graphs. A direct approach seems conceivable (Kohayakawa, Schacht)

22. Questions?