## Abstract

A cover of a hypergraph is a collection of edges whose union contains all vertices. Let H = (V, E) be a k-uniform, D-regular hypergraph on n vertices, in which no two vertices are contained in more than o(D/e^{2k}log D) edges as D tends to infinity. Our results include the fact that if k = o(log D), then there is a cover of (1+o(1))n/k edges, extending the known result that this holds for fixed k. On the other hand, if k ≥ 4 log D then there are k-uniform, D-regular hypergraphs on n vertices in which no two vertices are contained in more than one edge, and yet the smallest cover has at least ω((n/k) log (k /log D)) edges. Several extensions and variants are also obtained, as well as the following geometric application. The minimum number of lines required to separate n random points in the unit square is, almost surely, θ(n^{2/3}/(log n)^{1/3}).

Original language | English (US) |
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Pages (from-to) | 273-301 |

Number of pages | 29 |

Journal | Proceedings of the London Mathematical Society |

Volume | 86 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2003 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- General Mathematics