Asymmetric Ramsey Properties of Random Graphs involving Cliques

1. Asymmetric Ramsey Propertiesof 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.: 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 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 non-completegraphs(e.g., randomgraphs). • AvoidsomefixedgraphFother thanK`. • Avoid graph F1 in blue and F2 in red (asymmetric case). • Allow more colors.

4. 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 monotone increasing. •  The property R(F, k) has a threshold (Bollobás, Thomason, 1987). 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).

5. 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. • thisforcesthecopies of F to overlapsubstantially and makescoloringdifficult. • Order of magnitude of thresholddoesnotdepend on k (!)

6. R(L, R) Denote the family of all graphs that contain either a red copy of graph Lor a blue copy of graph Rin every edge coloringwith red and blue byR(L, R). Asymmetric Ramsey properties • Whathappensifwewant to avoiddifferentgraphsFi in different colorsi, 1·i·k? • Wefocus on thecasewithtwocolors.

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

8. The 0-Statement • Our proof is constructive: • We propose an algorithm that computes a valid coloring of Gn,pa.a.s. • All previous proofs were non-constructive. 0-Statement

9. The coloring algorithm • Algorithm proceeds in 2 phases: • remove edges from G one by one in some clever way • reinsert the edges 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 K4 nor a blue K5

10. The coloring algorithm • Advanced Idea: In Phase 2, before coloring the inserted edge, we may recolor an existing edge first. • Example:coloring without a red K4 nor a blue K5 (ctd.)

11. Sunflowers • Sunflowers: `-cliques, each edge of which is intersection with an r-clique. • Example:r = 5and`=4 • Only `-cliques that are the center of a sunflower contain no edge that can be recolored. • outerr-cliques may mutually overlap! • Def: `-clique isdangerous := `-clique iscenter of sunflower.

12. Thecoloringalgorithm • Basic idea: in Phase 1, successively removeedges which are not exactly the intersection of an `-clique with an r-clique. • these can be colored directly when reinserted in Phase 2.

13. Thecoloringalgorithm • Advanced idea: in Phase 1, successively remove edges which are not exactly the intersection of a dangerous`-clique with an r-clique. • these can be coloredpossibly after recoloring an existing edge when reinserted in Phase 2. • (recall: dangerous = center of sunflower = cannot guarantee that there is an edge that can be recolored) • The algorithm can be shown to remove all edges from Gn,p in Phase 1 a.a.s. unless `= 3.  Algorithm needs to be refined for triangles.

14. The trouble with triangles • If pbn-1/m2(K3, Kr),then G=Gn,pa.a.s. contains Kr+1. • Algorithm gets stuck in Phase 1 since every edge of Kr+1is the intersection of a dangerous K3and a Kr. • Even nastier substructures may appear. • Solution: • Determine those structures. • Color them separately at the beginning of phase 2. • Example:`= 3, r= 4

15. Lemma For pbn-1/m2(K`, Kr), the Coloring Algorithm terminates a.a.s. and produces a valid coloring of Gn,p. Proof sketch • To do: Show that a.a.sall 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 Gn,p. • If algorithm gets stuck on some graph G, every edge of G is contained in a dangerous `-clique (center of a sunflower). Using this property iteratively for the edges in the surrounding r-cliques (‘petals’ of the sunflower), we can build a subgraph of G(‘sunflower patch’) which is either too dense or too large to appear in Gn,p

16. Proof Sketch • Example:r = 5and`=4 • Main technicaldifficulty: handle overlappingpetals • Deterministic Lemma: If algorithm fails in Phase 1, then G contains • either a densesunflower patch (much overlap), • or a largesunflower patch (little overlap). • Probabilistic Lemma: With high probability, Gn,pcontains neither a dense nor a large sunflower patch.

17. Remarks • The generalization to asymmetric Ramsey properties with more than two colors is straightforward, since the conjectured threshold depends only on the two densest graphs Fi. • 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 sunflower petals • determine graphs which may remain after Phase 1

18. Questions?