## 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 non-empty 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 of 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-1dp+1 the piercing number is HD_{d}(p, q) = p- q+ 1 ; no tight bounds on 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 classes. The best-studied among them is the class of axis-parallel boxes in R^{d}, and specifically, axis-parallel rectangles in the plane. Wegner (Israel J Math 3:187–198, 1965) and (independently) Dol’nikov (Sibirsk Mat Ž 13(6):1272–1283, 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 classes with Helly number 2, even without assuming that the sets in the class 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, 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 class G of compact convex sets in R^{d} with Helly number 2 admits a (p, 2)-theorem with piercing number O(p^{2}^{d}^{-}^{1}) , and thus, satisfies HD _{G}(p, q) = p- q+ 1 , for a universal constant c.

Original language | English |
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Pages (from-to) | 821-847 |

Number of pages | 27 |

Journal | Discrete and Computational Geometry |

Volume | 63 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jun 2020 |

Externally published | Yes |

## Keywords

- (p,q)-Theorem
- Axis-parallel rectangles
- Convexity
- Hadwiger–Debrunner numbers
- Helly-type theorems