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 Kk. It is shown that for every fixed k and all n > n0(k), F (n, 2, k) = 2k−1 and F (n, 3, k) = 3k−1, where tk−1(n) is the maximum possible number of edges of a graph on n vertices with no Kk, (determined by Turán’s Theorem). The case r = 2 settles a conjecture of Yuster. On the other hand, for every fixed r > 3 and k > 2, the function F (n, r, k) is exponentially bigger than rk−1. The proofs are based on Szemerédi’s regularity lemma together with some additional tools in Extremal Graph Theory, and provide an example of a precise result proved by applying this lemma.