A probabilistic counting lemma for complete graphs

We prove the existence of many complete graphs in almost all sufficiently dense partitions obtained by an application of Szemerédi’s Regularity Lemma. More precisely, we consider the number of complete graphs K` on ` vertices in `-partite graphs where each partition class consists of n vertices and there is an ε-regular graph onm edges between any two partition classes. We show that for all β > 0, at most a β-fraction of graphs in this family contain less than the expected number of copies ofK` provided ε is sufficiently small andm ≥ Cn2−1/(`−1) for a constantC > 0 and n sufficiently large. This result is a counting version of a restricted version of a conjecture by Kohayakawa, Łuczak and Rödl [8] and has several implications for random graphs.