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 = cd(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 cd(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 c2(image, found), while the best known upper bound is O(image, found). In this paper we improve the lower bound significantly by showing that c2 (p, q) ≥ p1+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 c2 (p, 3). We then generalize the bound to c2(p, q) for any q ≥ 3.
|Number of pages||6|
|Journal||Acta Mathematica Universitatis Comenianae|
|State||Published - 2 Sep 2019|
ASJC Scopus subject areas
- Mathematics (all)