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.

Abstract. We study thresholds for extremal properties of random discrete structures.We determine the threshold for Szemerédi's theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we determine the threshold for Turán-type problems for random graphs and hypergraphs. In particular, we verify a conjecture of Kohayakawa, Łuczak, and Rödl for Turán-type problems in random graphs.Similar results were obtained independently by Conlon and Gowers.

Szemerédi's regularity lemma for graphs has proved to be a powerful tool with many subsequent applications. The objective of this paper is to extend the techniques developed by Nagle, Skokan, and the authors and obtain a stronger and more ‘user-friendly’ regularity lemma for hypergraphs.

In this paper we prove the following conjecture by Bollobás and Komlós: For every γ > 0 and integers r ≥ 1 and ∆, there exists β > 0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least ((r − 1)/r + γ )n and H is an r -chromatic graph with n vertices, bandwidth at most βn and maximum degree at most ∆, then G contains a copy of H .

We study sufficient ℓ-degree (1 ≤ ℓ < k) conditions for the appearance of perfect and nearly perfect matchings in k-uniform hypergraphs. In particular, we obtain a minimum vertex degree condition (ℓ = 1) for 3-uniform hypergraphs, which is approximately tight, by showing that every 3-uniform hypergraph on n vertices with minimum vertex degree at least (5/9 + o(1))`n 2ć ontains a perfect matching.