The number of edge colorings with no monochromatic cliques

Let F(n, r, k) denote the maximum possible number of distinct edge-colorings of a simple graph on n vertices with r colors which contain no monochromatic copy of K_k. It is shown that for every fixed k and all n > n₀(k), F(n, 2, k) = 2^tk-1(n) and F(n, 3, k) = 3^tk-1(n), where t_k-1(n) is the maximum possible number of edges of a graph on n vertices with no K_k (determined by Turán's theorem). The case r = 2 settles an old conjecture of Erdos and Rothschild, which was also independently raised later by Yuster. On the other hand, for every fixed r > 3 and k > 2, the function F(n, r, k) is exponentially bigger than r^tk-1(n). The proofs are based on Szemerédi's regularity lemma together with some additional tools in extremal graph theory, and provide one of the rare examples of a precise result proved by applying this lemma.