A new lower bound on Hadwiger-Debrunner numbers in the plane

Abstract

A family of sets ${\cal F}$ is said to satisfy the (p, q) property if among any p sets in ${\cal F}$ 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 $O({p^{(1.5 + \delta)(1 + {1 \over {q - 2}})}})$ (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 $H\,{D_2}(p,q) = \Omega \left({{p \over q}\log \left({{p \over q}} \right)} \right)$, 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.