Supersaturation for hereditary properties

Let F be a collection of r-uniform hypergraphs, and let 0 < p < 1. It is known that there exists c = c(p,F) such that the probability of a random r-graph in G(n, p) not containing an induced subgraph from F is 2(−c+o(1))( n r). Let each r-graph in F have at least t vertices. We show that in fact for every > 0, there exists δ = δ( , p,F) > 0 such that the probability of a random r-graph in G(n, p) containing less than δn induced subgraphs each lying in F is at most 2(−c+ )( n r). This statement is an analogue for hereditary properties of the supersaturation theorem of Erdős and Simonovits. In our applications we answer a question of Bollobás and Nikiforov.