Regularity Lemma for k-uniform hypergraphs

Szemerédi’s Regularity Lemma proved to be a very powerful tool in extremal graph theory with a large number of applications. Chung [Regularity lemmas for hypergraphs and quasi-randomness, Random Structures and Algorithms 2 (1991), 241–252], Frankl and Rödl [The uniformity lemma for hypergraphs, Graphs and Combinatorics 8 (1992), 309–312, Extremal problems on set systems, Random Structures and Algorithms 20 (2002), 131–164] considered several extensions of Szemerédi’s Regularity Lemma to hypergraphs. In particular, [Extremal problems on set systems, Random Structures and Algorithms 20 (2002), 131–164] contains a regularity lemma for 3uniform hypergraphs that was applied to a number of problems. In this paper, we present a generalization of this regularity lemma to k-uniform hypergraphs. Similar results were independently and alternatively obtained by W. T. Gowers.