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 a minimum integer c = HD_d(p, q), such that any finite family of convex sets in ℝ^d 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 HD_d(p, q), known as ‘the Hadwiger-Debrunner numbers’, is still a major open problem in discrete and computational geometry. The best known upper bound on the Hadwiger-Debrunner numbers in the plane is O(p(1.5+δ)(1+1q−2)) (for any δ > 0 and p ≥ q ≥ q₀(δ)), obtained by combining the results of Keller, Smorodinsky and Tardos (2017) and of Rubin (2018). The best lower bound is HD2(p,q)=Ω(pqlog(pq)), obtained by Bukh, Matoušek and Nivasch more than 10 years ago. In this paper we improve the lower bound significantly by showing that HD₂(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 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.