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Hypergraphs, Quasi-randomness, and Conditions for Regularity

✍ Scribed by Yoshiharu Kohayakawa; Vojtěch Rödl; Jozef Skokan

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
286 KB
Volume
97
Category
Article
ISSN
0097-3165

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✦ Synopsis

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 work may be viewed as a continuation of the work of Chung and Graham, although our proof techniques are different in certain important parts.

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