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 ) i s O ˜ ( p d 2 + d ) . We present several improved bounds: (i) For any q ≥ d + 1 , H D d ( p , d ) = O ˜ ( p d ( q − 1 q − d ) ) . (ii) For q ≥ log p , H D d ( p , q ) = O ˜ ( p + ( p / q ) d ) . (iii) For every ϵ > 0 there exists a p 0 = p 0 (ϵ) such that for every p ≥ p 0 and for every q ≥ p d − 1 d + ∈ 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.