Hypergraphs, Quasi-randomness, and Conditions for Regularity

Haviland 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 w...

• Type: Journal
• Year: 2002
• Where: JCT
• Authors: Yoshiharu Kohayakawa, Vojtech Rödl, Jozef Skokan
• Keywords: Disparate Properties, JCT 2002, Quasi-random Hypergraphs, Random-like Hypergraphs