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 Rd 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 \(H{D_d}(p,d + 1)is\tilde O({p^{{d^2} + d}})\).We present several improved bounds: (i) For any \(q \geqslant d + 1,H{D_d}(p,d) = \tilde O({p^{d(\frac{{q - 1}}{{q - d}})}})\). (ii) For q ≥ log p, \(H{D_d}(p,q) = \tilde O(p + {(p/q)^d})\). (iii) For every ? > 0 there exists a p₀ = p₀(?) such that for every p ≥ p₀ and for every \(q \geqslant {p^{\frac{{d - 1}}{d} + \in }}\) 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.