### Abstract

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices is j-independent if its intersection with any edge has size strictly less than j. The Kruskal-Katona theorem implies that in an r-uniform hypergraph with a fixed size and order, the hypergraph with the most r-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number of j-independent sets in an r-uniform hypergraph.

Original language | English |
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Pages (from-to) | 9-20 |

Number of pages | 12 |

Journal | Combinatorics Probability and Computing |

Volume | 22 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2013 |

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### Cite this

*Combinatorics Probability and Computing*,

*22*(1), 9-20. https://doi.org/10.1017/S0963548312000454