Weak hypergraph regularity and linear hypergraphs

  • Authors:

  • Yoshiharu Kohayakawa;Brendan Nagle;Vojtěch Rödl;Mathias Schacht

  • Affiliations:

  • Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090 São Paulo, Brazil;Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA;Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA;Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2010

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Abstract

We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any k-uniform hypergraph H of positive uniform density contains all linear k-uniform hypergraphs of a given size. More precisely, we show that for all integers @?=k=2 and every d0 there exists @r0 for which the following holds: if H is a sufficiently large k-uniform hypergraph with the property that the density of H induced on every vertex subset of size @rn is at least d, then H contains every linear k-uniform hypergraph F with @? vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph @e-regularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems.