Extremal problems on set systems

Abstract

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.

Integer and fractional packings in dense 3‐uniform hypergraphs

“…In this section we describe the regularity lemma for 3-uniform hypergraphs, due to Frankl and Rödl [6]. In the original Regularity Lemma of Szemerédi for graphs, it was shown that the vertex set of any large enough graph G can be partitioned into a bounded number of classes, such that the following holds: For almost every pair of vertex classes V i and V j , the bipartite subgraph of G induced by V i and V j is "very uniform," that is, its edges are distributed in much the same way as one would expect in a random bipartite graph.…”

“…If the regularity condition fails to be satisfied for any ␣, we say that Ᏼ is (␦, r)-irregular with respect to G. This implies in particular that if Ᏼ is a dense hypergraph, then most of the triples of Ᏼ belong to (␦, r)-regular triads of the partition ᏼ. We now state the Regularity Lemma of [6]. …”

“…We say that the partition ᏼ respects the partition ᏽ if every vertex class of ᏼ is entirely contained in some vertex class of ᏽ. Then the following can be proved with a minor alteration of the proof of [6]. …”

“…The proof of Theorem 1.1 depends on a generalization of Szemerédi's Lemma to hypergraphs, which is due to Frankl and Rödl [6] (see Section 2). Our plan for proving that 0 (Ᏼ) is closely approximated by * 0 (Ᏼ) for a given hypergraph Ᏼ will be as follows.…”

“…In particular, we prove a result that we call the Extension Theorem. This states that if a k-partite 3-uniform hypergraph is regular [in the sense of the hypergraph regularity lemma of Frankl and Rödl (2002)], then almost every triple is in about the same number of copies of K k (3) (the complete 3-uniform hypergraph with k vertices). © 2003 Wiley Periodicals, Inc.…”