Improved bounds on the Hadwiger–Debrunner numbers

Let HD_d(p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in ℝ^d which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger– Debrunner conjecture, Alon and Kleitman proved that HD_d(p, q) exists for all p ≥ q ≥ d + 1. Specifically, they prove that HD_d(p, d+1) is Õ(image found). We present several improved bounds: (i) For any q ≥ d+1, HDd(p, q) = Õ (image found). (ii) For q ≥ log p, HD_d(p, q) = Õ (p+(p/q)^d). (iii) For every ε > 0 there exists a p₀ = p₀(ε) such that for every p ≥ p₀ and for every (image found) we have p − q + 1 ≤ HD_d(p, q) ≤ p − q + 2. The latter is the first near tight estimate of HD_d(p, q) for an extended range of values of (p, q) since the 1957 Hadwiger–Debrunner theorem. We also prove a (p, 2)-theorem for families in ℝ² with union complexity below a specific quadratic bound.