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 an integer c = HDd(p,q), such that any finite family of convex sets in ℝd 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 HDd(p,q), known as ‘the Hadwiger-Debrunner numbers', is still a major open problem in discrete and computational geometry. The best currently known upper bound on the Hadwiger-Debrunner numbers in the plane is (for any σ > 0 and p ≥ q ≥ q0(δ)), obtained by combining results of Keller, Smorodinsky, and Tardos (SODA 2017) and of Rubin (FOCS 2018). The best lower bound is , obtained by Bukh, Matoušek and Nivasch more than 10 years ago. In this paper we improve the lower bound significantly by showing that . 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 Erdős on points in general position in the plane. We use a novel construction for Erdős' problem, obtained recently by Balogh and Solymosi using the hypergraph container method, to get the lower bound on HD2(p,3). We then generalize the bound to HD2(p, q) for q ≥ 3.