The counting lemma for regular k-uniform hypergraphs

Szemeredi'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 e-partite graph with V (G) = V1 ∪ … ∪ Ve and sVis = n for all i ∈ [e], and all pairs (Vi, Vj) are e-regular of density d for 1 ≤ i ≤ j ≤ e and e L d, then G contains $(1\pm f_{\ell}(\varepsilon))d^{\ell \choose 2}\times n^{\ell}$ cliques Ke, where fe(e) → 0 as e → 0.Recently, Rodl and Skokan generalized Szemeredi's Regularity Lemma from graphs to k-uniform hypergraphs for arbitrary k ≥ 2. In this paper we prove a Counting Lemma accompanying the Rodl–Skokan hypergraph Regularity Lemma. Similar results were independently obtained by Gowers.Such results give combinatorial proofs to the density result of Szemeredi and some of the density theorems of Furstenberg and Katznelson. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006