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For a family F k = k 1 k 2 k t of k-uniform hypergraphs let ex n F k denote the maximum number of k-tuples which a k-uniform hypergraph on n vertices may have, while not containing any member of F k . Let r k n denote the maximum cardinality of a set of integers Z ⊂ n , where Z contains no arithmetic progression of length k. For any k ≥ 3 we introduce families F k = k 1 k 2 and prove thatholds. We conjecture that ex n F k = o n k−1 holds. If true, this would imply a celebrated result of Szemerédi stating that r k n = o n . By an earlier result o Ruzsa and Szemerédi, our conjecture is known to be true for k = 3. The main objective of this article is to verify the conjecture for k = 4. We also consider some related problems.

Abstract. Let H be a fixed graph of chromatic number r. It is shown that the number of graphs on n n 1 vertices and not containing H as a subgraph is 2(2)(1-,-~ -÷°~1~). Let h,(n) denote the maximum number of edges in an r-uniform hypergraph on n vertices and in which the union of any three edges has size greater than 3r -3. It is shown that h,(n) = o(n 2) although for every fixed c < 2 one has lim~ h,(n)/n ~ = oo.

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In this paper we study degree conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erdős on estimating the maximum number of edges in a hypergraph when the (fractional) matching number is given, which we are able to solve in some special cases using probabilistic techniques. Based on these results, we obtain some general theorems on the minimum d-degree ensuring the existence of perfect (fractional) matchings. In particular, we asymptotically determine the minimum vertex degree which guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also discuss an application to a problem of finding an optimal data allocation in a distributed storage system.