Abstract. Szemerédi's Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an -partite graph with V (G) = V 1 ∪ · · · ∪ V and |V i | = n for all i ∈ [ ], and all pairs (V i , V j ) are ε-regular of density d forRecently, V. Rödl and J. Skokan generalized Szemerédi's Regularity Lemma from graphs to k-uniform hypergraphs for arbitrary k ≥ 2. In this paper we prove a Counting Lemma accompanying the Rödl-Skokan hypergraph Regularity Lemma. Similar results were independently and alternatively obtained by W. T. Gowers.It is known that such results give combinatorial proofs to the density result of E. Szemerédi and some of the density theorems of H. Furstenberg and Y. Katznelson.

We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any k-uniform hypergraph H of positive uniform density contains all linear k-uniform hypergraphs of a given size. More precisely, we show that for all integers k 2 and every d > 0 there exists > 0 for which the following holds: if H is a sufficiently large k-uniform hypergraph with the property that the density of H induced on every vertex subset of size n is at least d, then H contains every linear k-uniform hypergraph F with vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph ε-regularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems.

No abstract

Szemerédi's Regularity Lemma is a well-known and powerful tool in modern graph theory. This result led to a number of interesting applications, particularly in extremal graph theory. A regularity lemma for 3-uniform hypergraphs developed by Frankl and Rödl [8] allows some of the Szemerédi Regularity Lemma graph applications to be extended to hypergraphs. An important development regarding Szemerédi's Lemma showed the equivalence between the property of ⑀-regularity of a bipartite graph G and an easily verifiable property concerning the neighborhoods of its vertices (Alon et al. [1]; cf. [6]). This characterization of ⑀-regularity led to an algorithmic version of Szemerédi's lemma [1]. Similar problems were also considered for hypergraphs. In [2], [9], [13], and [18], various descriptions of quasi-randomness of k-uniform hypergraphs were given. As in [1], the goal of this paper is to find easily verifiable conditions for the hypergraph regularity provided by [8]. The hypergraph regularity of [8] renders quasi-random "blocks of hyperedges" which are very sparse. This situation leads to technical difficulties in its application. Moreover, as we show in this paper, some easily verifiable conditions analogous to those considered in [2] and [18] fail to be true in the setting of [8]. However, we are able to find some necessary and sufficient 293conditions for this hypergraph regularity. These conditions enable us to design an algorithmic version of a hypergraph regularity lemma in [8]. This algorithmic version is presented by the authors in [5].