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.
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 define a perfect matching in a k-uniform hypergraph H on n vertices as a set of n/k disjoint edges. Let δ k−1 (H) be the largest integer d such that every (k − 1)-element set of vertices of H belongs to at least d edges of H.In this paper we study the relation between δ k−1 (H) and the presence of a perfect matching in H for k 3. Let t(k, n) be the smallest integer t such that every k-uniform hypergraph on n vertices and with δ k−1 (H) t contains a perfect matching.For large n divisible by k, we completely determine the values of t(k, n), which turn out to be very close to n/2 − k. For example, if k is odd and n is large and even, then t(k, n) = n/2 − k + 2. In contrast, for n not divisible by k, we show that t(k, n) ∼ n/k.In the proofs we employ a newly developed "absorbing" technique, which has a potential to be applicable in a more general context of establishing existence of spanning subgraphs of graphs and hypergraphs.
(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.
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