A generalized Turán problem in random graphs

We study a generalization of the Turán problem in random graphs. Given graphs T and H, let ex(G(n,p),T,H) be the largest number of copies of T in an H-free subgraph of G(n,p). We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every H and every 2-balanced T. Our results in the case when m₂(H) > m₂(T) are a natural generalization of the Erdős-Stone theorem for G(n,p), proved several years ago by Conlon-Gowers and Schacht; the case T = K_m was previously resolved by Alon, Kostochka, and Shikhelman. The case when m₂(H) ≤ m₂(T) exhibits a more complex behavior. Here, the location(s) of the (possibly multiple) threshold(s) are determined by densities of various coverings of H with copies of T and the typical value(s) of ex(G(n,p),T,H) are given by solutions to deterministic hypergraph Turán-type problems that we are unable to solve in full generality.