The Regularity Method for Sparse Graphs and Hypergraphs

The Regularity Lemma of E. Szemeredi for graphs asserts that every dense graph can be decomposed into relatively few random-like subgraphs. This random-like behavior enables one to find and enumerate subgraphs of a given isomorphism type. This observation is called Counting Lemma. The interplay of Szemeredi's Regularity Lemma and the Counting Lemma has had a series of successes in Discrete Mathematics. In recent years the Regularity Method was partly expanded to new types of discrete structures: sparse graphs and k-uniform hypergraphs. In particular, analogues of the Regularity Lemma for these combinatorial objects were established. In the proposed program, we seek a deeper understanding of the random-like behavior guaranteed by those Regularity Lemmas. Furthermore, we focus on applications of these novel techniques in the area of extremal combinatorics.