## Abstract

The independence number α(H) 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 H_{n} is an n-vertex (r+1)-uniform hypergraph in which every r-element set is contained in at most d edges, where 0<d<n/(logn)3r2, then α(Hn)≥cr(ndlog<>nd)1/r where cr>0 satisfies cr~r/e as r→∞. The value of c_{r} 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 r≥2 and all values of d∈ℕ, there are infinitely many H_{n} such that α(Hn)≤br(ndlog<>nd)1/r where br>0 depends only on r. In addition, for many values of d we show br~cr as r→∞, so the result is almost sharp for large r. We give an application to hypergraph Ramsey numbers involving independent neighborhoods.

Original language | English (US) |
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Pages (from-to) | 224-239 |

Number of pages | 16 |

Journal | Random Structures and Algorithms |

Volume | 44 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2014 |

## Keywords

- Hypergraphs
- Independent sets
- Steiner systems

## ASJC Scopus subject areas

- Software
- General Mathematics
- Computer Graphics and Computer-Aided Design
- Applied Mathematics