## Abstract

A family F of sets is said to satisfy the (p, q)-property if among any p sets of F some q have a nonempty intersection. The celebrated (p, q)-theorem of Alon and Kleitman asserts that any family of compact convex sets in ℝ^{d} that satisfies the (p, q)-property for some q ≥ d + 1, can be pierced by a fixed number (independent on the size of the family) f_{d}(p, q) of points. The minimum such piercing number is denoted by HD_{d}(p, q). Already in 1957, Hadwiger and Debrunner showed that whenever q > d-1/d p + 1 the piercing number is HD_{d}(p, q) = p - q + 1; no exact values of HD_{d}(p, q) were found ever since. While for an arbitrary family of compact convex sets in ℝ^{d}, d ≥ 2, a (p, 2)-property does not imply a bounded piercing number, such bounds were proved for numerous specific families. The best-studied among them is axis-parallel boxes in ℝ^{d}, and specifically, axis-parallel rectangles in the plane. Wegner (1965) and (independently) Dol'nikov (1972) used a (p, 2)-theorem for axis-parallel rectangles to show that HD_{rect}(p, q) = p - q + 1 holds for all q > √2p. These are the only values of q for which HD_{rect}(p, q) is known exactly. In this paper we present a general method which allows using a (p, 2)-theorem as a bootstrapping to obtain a tight (p, q)-theorem, for families with Helly number 2, even without assuming that the sets in the family are convex or compact. To demonstrate the strength of this method, we show that HD_{d-box}(p, q) = p - q + 1 holds for all q > c'log^{d-1} p, and in particular, HD_{rect}(p, q) = p - q + 1 holds for all q ≥ 7 log_{2} p (compared to q ≥ √2p, obtained by Wegner and Dol'nikov more than 40 years ago). In addition, for several classes of families, we present improved (p, 2)-theorems, some of which can be used as a bootstrapping to obtain tight (p, q)-theorems. In particular, we show that any family F of compact convex sets in ℝ^{d} with Helly number 2 admits a (p, 2)-theorem with piercing number O(p^{2d-1}), and thus, satisfies HD_{F}(p, q) = p - q + 1 for all q > cp^{1-1/2d-1}, for a universal constant c.

Original language | English |
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Title of host publication | 34th International Symposium on Computational Geometry, SoCG 2018 |

Editors | Csaba D. Toth, Bettina Speckmann |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 511-5114 |

Number of pages | 4604 |

ISBN (Electronic) | 9783959770668 |

DOIs | |

State | Published - 1 Jun 2018 |

Externally published | Yes |

Event | 34th International Symposium on Computational Geometry, SoCG 2018 - Budapest, Hungary Duration: 11 Jun 2018 → 14 Jun 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 99 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 34th International Symposium on Computational Geometry, SoCG 2018 |
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Country/Territory | Hungary |

City | Budapest |

Period | 11/06/18 → 14/06/18 |

## Keywords

- (P,Q)-Theorem
- (p,2)-theorem
- Axis-parallel rectangles
- Convexity
- Transversals