A remark on the number of edge colorings of graphs

Fix a 2-coloring H_k+1 of the edges of a complete graph K_k+1. Let C(n, H_k+1) denote the maximum possible number of distinct edge-colorings of a simple graph on n vertices with two colors, which contain no copy of K_k+1 colored exactly as H_k+1. It is shown that for every fixed k and all n > n₀(k), if in the colored graph H_k+1 both colors were used, then C (n, H_k+1) = 2^tk(n), where t_k(n) is the maximum possible number of edges of a graph on n vertices containing no K_k+1. The proofs are based on Szemerédi's Regularity Lemma together with the Simonovits Stability Theorem, and provide one of the growing number of examples of a precise result proved by applying the Regularity Lemma.