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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 This paper has two parts. In the first part we present several improved bounds on HD d (p, q). In particular, we obtain the first near tight estimate of HD d (p, q) for an extended range of values of (p, q) since the 1957doi:10.1137/1.9781611974782.148 dblp:conf/soda/KellerST17 fatcat:bhcytgudnfblzeibvhlmcvotp4