The Erd\H{o}s-Hajnal conjecture for three colors and multiple forbidden patterns

Abstract

Erd\H{o}s and Szekeres's quantitative version of Ramsey's theorem asserts that any complete graph on n vertices that is edge-colored with two colors has a monochromatic clique on at least 1/2log(n) vertices. The famous Erd\H{o}s-Hajnal conjecture asserts that forbidding fixed color patterns ensures larger monochromatic cliques. Specifically, it claims that for any fixed integer k and any clique K on k vertices edge-colored with two colors, there is a positive constant a such that in any complete n-vertex graph edge-colored with two colors that does not contain a copy of K, there is a monochromatic clique on at least n^a vertices. We consider edge-colorings with three colors. For a family H of triangles, each colored with colors from {r, b, y}, Forb(n,H) denotes a family of edge-colorings of the complete n-vertex graph using colors from {r, b, y} and containing none of the colorings from H. Let h_2(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 h_2(n, H) for all families H consisting of at most three triangles. For most of them, our bounds are asymptotically tight. This extends a result of Fox, Grinshpun, and Pach, who determined h_2(n, H) for H consisting of a rainbow triangle, and confirms the multicolor Erd\H{o}s-Hajnal conjecture for these sets of patterns.