Hypergraph regularity and quasi-randomness

  • Authors:

  • Brendan Nagle;Annika Poerschke;Vojtěch Rödl;Mathias Schacht

  • Affiliations:

  • University of South Florida, Tampa, FL;Emory University, Atlanta, GA;Emory University, Atlanta, GA;Humboldt-Universität zu Berlin, Berlin, Germany

  • Venue:
  • SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
  • Year:
  • 2009

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Abstract

Thomason and Chung, Graham, and Wilson were the first to systematically study quasi-random graphs and hypergraphs, and proved that several properties of random graphs imply each other in a deterministic sense. Their concepts of quasi-randomness match the notion of ε-regularity from the earlier Szemerédi regularity lemma. In contrast, there exists no "natural" hypergraph regularity lemma matching the notions of quasi-random hypergraphs considered by those authors. We study several notions of quasi-randomness for 3-uniform hypergraphs which correspond to the regularity lemmas of Frankl and Rödl, Gowers and Haxell, Nagle and Rödl. We establish an equivalence among the three notions of regularity of these lemmas. Since the regularity lemma of Haxell et al. is algorithmic, we obtain algorithmic versions of the lemmas of Frankl-Rödl (a special case thereof) and Gowers as corollaries. As a further corollary, we obtain that the special case of the Frankl-Rödl lemma (which we can make algorithmic) admits a corresponding counting lemma. (This corollary follows by the equivalences and that the regularity lemma of Gowers or that of Haxell et al. admits a counting lemma.)