## Abstract

The Hadwiger–Debrunner number HD_{d}(p,q) is the minimal size of a piercing set that can always be guaranteed for a family of compact convex sets in R^{d} that satisfies the (p,q) property. Hadwiger and Debrunner showed that HD_{d}(p,q)≥p−q+1 for all q, and equality is attained for q>[Formula presented]p+1. Almost tight upper bounds for HD_{d}(p,q) for a ‘sufficiently large’ q were obtained recently using an enhancement of the celebrated Alon–Kleitman theorem, but no sharp upper bounds for a general q are known. In [9], Montejano and Soberón defined a refinement of the (p,q) property: F satisfies the (p,q)_{r} property if among any p elements of F, at least r of the q-tuples intersect. They showed that HD_{d}(p,q)_{r}≤p−q+1 holds for all r>(pq)−(p−d+1q−d+1); however, this is far from being tight. In this paper we present improved asymptotic upper bounds on HD_{d}(p,q)_{r} which hold when only a tiny portion of the q-tuples intersect. In particular, we show that for p,q sufficiently large, HD_{d}(p,q)_{r}≤p−q+1 holds with r=p^{−q/2d}⋅(pq). Our bound misses the known lower bound for the same piercing number by a factor of less than pq^{d}. Our results use Kalai's Upper Bound Theorem for convex sets, along with the Hadwiger–Debrunner theorem and the recent improved upper bound on HD_{d}(p,q) mentioned above.

Original language | English |
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Pages (from-to) | 11-18 |

Number of pages | 8 |

Journal | Computational Geometry: Theory and Applications |

Volume | 72 |

DOIs | |

State | Published - Jun 2018 |

Externally published | Yes |

## Keywords

- (p,q)-Theorem
- Convexity
- Hadwiger–Debrunner number
- Helly-type theorems
- Upper bound theorem

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