The Erdős-Hajnal conjecture for rainbow triangles

We prove that every 3-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Ω ( n log n ) which uses at most two colors, and this bound is tight up to a constant factor. This verifies a conjecture of Hajnal which is a case of the multicolor generalization of the well-known Erdős-Hajnal conjecture. We further establish a generalization of this result. For fixed positive integers s and r with s ≤ r, we determine a constant cr,s such that the following holds. Every r-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Ω ( nr(r−1)/s(s−1)(log n)r,s ) which uses at most s colors, and this bound is tight apart from the implied constant factor. The proof of the lower bound utilizes Gallai’s classification of rainbow-triangle free edge-colorings of the complete graph, a new weighted extension of Ramsey’s theorem, and a discrepancy inequality in edge-weighted graphs. The proof of the upper bound uses Erdős’ lower bound on Ramsey numbers by considering lexicographic products of 2-edge-colorings of complete graphs without large monochromatic cliques.