General deletion lemmas via the Harris inequality

We present a new approach to the deletion method that is based on the Harris inequality. We obtain deletion lemmas that are similar in spirit to those of Rödl and Ruciński, but hold for arbitrary decreasing properties. That is, we show that under appropriate conditions, with very high (‘Janson-like’) probability it suffices to delete a small fraction of the edges of a random graph Gn,p to ensure that a given decreasing property holds. We also obtain stronger results for decreasing properties that only depend on edges involved in copies of some given graph H. As an application of our methods, we present a new deletion lemma that concerns local subgraph counts, i.e., the number of copies of H each edge or vertex of Gn,p is contained in.