Hypergraphs, quasi-randomness, and conditions for regularity

  • Authors:

  • Yoshiharu Kohayakawa;Vojtech Rödl;Jozef Skokan

  • Affiliations:

  • Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-900 São Paulo, Brazil;Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia;Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61820, and Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2002

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Abstract

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 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.