Graphs with Many Edge-Colorings Such That Complete Graphs Are Rainbow

We consider a version of the Erdős–Rothschild problem for families of graph patterns. For any fixed k ≥ 3, let r 0 ( k ) be the largest integer such that the following holds for all 2 ≤ r ≤ r 0 ( k ) and all sufficiently large n: The Turán graph T k − 1 ( n ) is the unique n-vertex graph G with the maximum number of r-edge-colorings such that the edge set of any copy of K k in G is rainbow. We use the regularity lemma of Szemerédi and linear programming to obtain a lower bound on the value of r 0 ( k ). For a more general family P of patterns of K k, we also prove that, in order to show that the Turán graph T k − 1 ( n ) maximizes the number of P-free r-edge-colorings over n-vertex graphs, it suffices to prove a related stability result.