Maximum number of r-edge-colorings such that all copies of Kk are rainbow

We consider a version of the Erdős-Rothschild problem for families graph patterns. For any fixed k ≥ 3, let r0(k) be largest integer such that following holds all 2 ≤ r and sufficiently large n: The Turán Tk-1(n) is unique n-vertex G with maximum number r-edge-colorings edge set copy Kk in rainbow. use regularity lemma Szemerédi linear programming to obtain lower bound on value r0(k). more general family P patterns Kk, we also prove that, order show maximizes P-free over graphs, it suffices related stability result.