Generalizations of the Ruzsa–Szemerédi and rainbow Turán problems for cliques

Abstract Considering a natural generalization of the Ruzsa–Szemerédi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing n^re^-O ({n) }=n^r-o (1) copies of K s such that any K r is contained in at most one K s. We also give bounds for the generalized rainbow Turán problem ex (n, H, rainbow - F) when F is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on n vertices with n^r-1-o (1) copies of K r such that no K r is rainbow.