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 We present several improved bounds: (iii) For every ǫ > 0 there exists a p 0 = p 0 (ǫ) such that for every p ≥ p 0 and for every q ≥ p d−1 d +ǫ we have: p − q + 1 ≤ HD d (p, q) ≤ p − q + 2. The latter is the first near tight

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... mate 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. Based on this, we introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying this property.