The Erdős-Hajnal conjecture for three colors and triangles

Abstract

Erdős and Szekeres's quantitative version of Ramsey's theorem asserts that in every 2-edge-coloring of Kn there is a monochromatic clique on at least 12log⁡n vertices. The famous Erdős-Hajnal conjecture claims that forbidding fixed colorings on subgraphs ensures much larger monochromatic cliques. Specifically, the most general, multi-color version of the conjecture states that for any fixed integers k,s and any s-edge-coloring c of Kk, there exists ε>0 such that in any s-edge-coloring of Kn that avoids c there is a clique on at least nε vertices, using at most s−1 colors. The conjecture is open in general, though a few partial results are known.Here, we focus on quantitative aspects of this conjecture in the case when k=3 and s=3, and when there are several forbidden subgraph colorings. More precisely, for a family H of triangles, each edge-colored with colors from {r,b,y}, Forb(n,H) denotes the family of edge-colorings of Kn using colors from {r,b,y} and containing none of the colorings from H. Let h2(n,H) be the maximum q such that any coloring from Forb(n,H) has a clique on at least q vertices using at most two colors. We provide bounds on h2(n,H) for all families H consisting of at most three triangles. For most of them, our bounds are asymptotically tight. This, in particular, extends a result of Fox, Grinshpun, and Pach, who determined h2(n,H) for H consisting of a rainbow triangle. In addition, we prove that for some H, h2(n,H) corresponds to certain classical Ramsey numbers, smallest independence number in graphs of given odd girth, or some other natural graph theoretic parameters.