Supersaturation for Subgraph Counts

The classical extremal problem is that of computing the maximum number of edges in an F-free graph. In particular, Turán’s theorem entirely resolves the case where F= K_r+1. Later results, known as supersaturation theorems, proved that in a graph containing more edges than the extremal number, there must also be many copies of K_r+1. Alon and Shikhelman introduced a broader class of extremal problems, asking for the maximum number of copies of a graph T in an F-free graph (so that T= K₂ is the classical extremal number). In this paper we determine some of these generalized extremal numbers when T and F are stars or cliques and prove some supersaturation results for them.