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 ϵ > 0 , there exists δ = δ ( ϵ , p , F ) > 0 such that the probability of a random r -graph in G ( n , p ) containing fewer than δ n t induced subgraphs each lying in F is at most 2 ( − c + ϵ ) n r . tatement 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.