A beautiful result of Szemerédi on the asymptotic structure of graphs is his regularity lemma. Roughly speaking, his result tells us that any large graph may be written as a union of a bounded number of induced, random looking bipartite graphs (the so called ε-regular pairs). Many applications of the regularity lemma are based on the following fact, often referred to as the counting lemma: Let G be an s-partite graph with vertex partition , where |Vi|=m for all i and all pairs (Vi,Vj) are ε-regular of density d. Then G contains  cliques Ks, where f(ε) → 0 as ε → 0. The combined application of the regularity lemma followed by the counting lemma is now often called the regularity method. In recent years, considerable advances have occurred in the applications of the regularity method, of which we mention two:
(i) the regularity lemma and the counting lemma have been generalized to the hypergraph setting and
(ii) the case of sparse graphs is now much better understood.
The developments in (i) are due, independently, to W.T. Gowers and to V. Rödl and his co-authors.
In this talk, we shall focus on the hypergraph version of the regularity method, following a series of works by P. Frankl, B. Nagle, V. Rödl, M. Schacht, and J. Skokan.