For two fixed graphs T and H, a positive integer n and a real number p in [0,1] let ex(G(n,p),T,H) be the random variable counting the maximum number of copies of T in an H-free subgraph of the random graph G(n,p). In this talk we discuss this variable, its phase transition as a function of p and its connection to the deterministic function counting the maximum number of copies of T in an H-free graph on n vertices.

Based on joint works with N. Alon, A. Kostochka and W. Samotij.