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 . We present several improved bounds: (i) For any . (ii) For q >= log p, . (iii) For every I mu > 0 there exists a p (0) = p (0)(I mu) such that for every p >= p (0) and for every 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 R (2) with union complexity below a specific quadratic bound.