### Abstract

A family of sets has the (p, q) property if among any p members of the family some q have a nonempty intersection. The authors have proved that for every p ≥ q ≥ d + 1 there is a c = c(p, q, d) < ∞ such that for every family F 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 F, thus settling an old problem of Hadwiger and Debrunner. Here we present a purely combinatorial proof of this result.

Original language | English (US) |
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Article number | R1 |

Pages (from-to) | 1-8 |

Number of pages | 8 |

Journal | Electronic Journal of Combinatorics |

Volume | 4 |

Issue number | 2 R |

State | Published - Dec 1 1997 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

Alon, N., & Kleitman, D. J. (1997). A purely combinatorial proof of the Hadwiger Debrunner (p, q) conjecture.

*Electronic Journal of Combinatorics*,*4*(2 R), 1-8. [R1].