A new lower bound on hadwiger-debrunner numbers in the plane

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 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.