A new lower bound on Hadwiger-Debrunner numbers in the plane
share
A set family 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 R^d that satisfies the (p,q) property, can be pierced by at most c points. In their celebrated (p,q) theorem from 1992, Alon and Kleitman proved the conjecture but did not obtain effective bounds on c_d(p,q), called `the Hadwiger-Debrunner numbers'. Ever since, obtaining such bounds is a major open problem in convexity theory. The best currently known asymptotic lower bound on the Hadwiger-Debrunner numbers in the plane is c_2(p,q) = Ω( p/q(p/q)). In this paper we present the significantly stronger lower bound c_2(p,q) ≥ p^1+Ω(1/q). This bound, obtained by an explicit family of lines, 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 the Erdős' problem, obtained recently by Balogh and Solymosi using the hypergraph container method, to get the lower bound on c_2(p,3). We then generalize the bound to c_2(p,q) for any q ≥ 3 with a construction based on a subset of the (2q-2)-dimensional grid.
READ FULL TEXT