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 ? ( n 1 / 3 log 2 ? 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 Erd?s-Hajnal conjecture. We further establish a generalization of this result. For fixed positive integers s and r with s ? r , we determine a constant c r , 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 ? ( n s ( s - 1 ) / r ( r - 1 ) ( log ? n ) c 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 Erd?s' lower bound on Ramsey numbers by considering lexicographic products of 2-edge-colorings of complete graphs without large monochromatic cliques.