## Abstract

A family of sets F is said to satisfy the (p, q)-property if among any p sets in F some q have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any p ≥ q ≥ d +1 there exists c = c_{d}(p, q), such that any family of compact convex sets in that satisfies the (p, q)-property can be pierced by at most c points. In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on c_{d}(p, q), known as the ‘the Hadwiger-Debrunner numbers,’ is still a major open problem in combinatorial geometry. The best currently known lower bound on the Hadwiger-Debrunner numbers in the plane is c_{2}(image, found), while the best known upper bound is O(image, found). In this paper we improve the lower bound significantly by showing that c_{2} (p, q) ≥ p^{1}+Ω (1/q). Furthermore, the bound is obtained by a family of lines and is tight for all families that have a bounded VC-dimension. Unlike previous bounds on the Hadwiger-Debrunner numbers, which mainly used the weak epsilon-net theorem, our bound stems from a surprising connection of the (p, q)-problem to an old problem of Erdos on points in general position in the plane. We use a novel construction for the Erdos’ problem, obtained recently by Balogh and Solymosi using the hypergraph container method, to get the lower bound on c_{2} (p, 3). We then generalize the bound to c_{2}(p, q) for any q ≥ 3.

Original language | English |
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Pages (from-to) | 855-860 |

Number of pages | 6 |

Journal | Acta Mathematica Universitatis Comenianae |

Volume | 88 |

Issue number | 3 |

State | Published - 2 Sep 2019 |

## ASJC Scopus subject areas

- General Mathematics