(2002), 131-164] contains a regularity lemma for 3-uniform 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 recently independently and alternatively obtained by W. T. Gowers.

Abstract. In this note we discuss several combinatorial problems that can be addressed by the Regularity Method for hypergraphs. Based on recent results of Nagle, Schacht and the authors, we give here solutions to these problems.In particular, we prove the following: Let K (k) t be the complete kuniform hypergraph on t vertices and suppose an n-vertex k-uniform hypergraph H contains only o(n t ) copies of K (k) t . Then one can delete o(n k ) edges of H to make it K (k) t -free. Similar results were recently obtained by W. T. Gowers.

A celebrated result of Chvátal, Rödl, Szemerédi and Trotter states (in slightly weakened form) that, for every natural number ∆, there is a constant r ∆ such that, for any connected n-vertex graph G with maximum degree ∆, the Ramsey number R(G, G) is at most r ∆ n, provided n is sufficiently large.In 1987, Burr made a strong conjecture implying that one may take r ∆ = ∆. However, Graham, Rödl and Ruciński showed, by taking G to be a suitable expander graph, that necessarily r ∆ > 2 c∆ for some constant c > 0. We show that the use of expanders is essential: if we impose the additional restriction that the bandwidth of G be at most some function β(n) = o(n), then R(G, G) ≤ (2χ(G) + 4)n ≤ (2∆ + 6)n, i.e., r ∆ = 2∆ + 6 suffices. On the other hand, we show that Burr's conjecture itself fails even for P k n , the kth power of a path P n . Brandt showed that for any c, if ∆ is sufficiently large, there are connected nvertex graphs G with ∆(G) ≤ ∆ but R(G, K 3 ) > cn. We show that, given ∆ and H, there are β > 0 and n 0 such that, if G is a connected graph on n ≥ n 0 vertices with maximum degree at most ∆ and bandwidth at most βn, then we have R(G, H) = (χ(H) − 1)(n − 1) + σ(H), where σ(H) is the smallest size of any part in any χ(H)-partition of H. We also show that the same conclusion holds without any restriction on the maximum degree of G if the bandwidth of G is at most ε(H) log n/ log log n.

dedicated to professors vera t. só s and andrás hajnal on the occasion of their 70th birthdaysHaviland and Thomason and Chung and Graham were the first to investigate systematically some properties of quasi-random hypergraphs. In particular, in a series of articles, Chung and Graham considered several quite disparate properties of random-like hypergraphs of density 1/2 and proved that they are in fact equivalent. The central concept in their work turned out to be the so called deviation of a hypergraph. They proved that having small deviation is equivalent to a variety of other properties that describe quasi-randomness. In this paper, we consider the concept of discrepancy for k-uniform hypergraphs with an arbitrary constant density d (0 < d < 1) and prove that the condition of having asymptotically vanishing discrepancy is equivalent to several other quasi-random properties of H, similar to the ones introduced by Chung and Graham. In particular, we prove that the correct ''spectrum'' of the s-vertex subhypergraphs is equivalent to quasi-randomness for any s \ 2k. Our work may be viewed as a continuation of the work of Chung and Graham, although our proof techniques are different in certain important parts.