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Keywords:

  • independent sets;
  • hypergraphs;
  • steiner systems

Abstract

The independence number inline image of a hypergraph H is the size of a largest set of vertices containing no edge of H. In this paper, we prove that if Hn is an n-vertex inline image-uniform hypergraph in which every r-element set is contained in at most d edges, where inline image, then

  • display math

where inline image satisfies inline image as inline image. The value of cr improves and generalizes several earlier results that all use a theorem of Ajtai, Komlós, Pintz, Spencer and Szemerédi (J Comb Theory Ser A 32 (1982), 321–335). Our relatively short proof extends a method due to Shearer (Random Struct Algorithms 7 (1995), 269–271) and Alon (Random Struct Algorithms 9 (1996), 271–278). The above statement is close to best possible, in the sense that for each inline image and all values of inline image, there are infinitely many Hn such that

  • display math

where inline image depends only on r. In addition, for many values of d we show inline image as inline image, so the result is almost sharp for large r. We give an application to hypergraph Ramsey numbers involving independent neighborhoods.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 224-239, 2014