### Abstract

A family of sets has the (p, q)property if among any p members of the family some q have a nonempty intersection. It is shown that for every p ≥ q ≥ d + 1 there is a c = c(p, q, d) < ∞ such that for every family J of compact, convex sets in R^{d} which has the (p, q) property there is a set of at most c points in R^{d} that intersects each member of J. This settles an old problem of Hadwiger and Debrunner.

Original language | English (US) |
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Pages (from-to) | 103-112 |

Number of pages | 10 |

Journal | Advances in Mathematics |

Volume | 96 |

Issue number | 1 |

DOIs | |

State | Published - Nov 1992 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Alon, N., & Kleitman, D. J. (1992). Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem.

*Advances in Mathematics*,*96*(1), 103-112. https://doi.org/10.1016/0001-8708(92)90052-M