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 1957

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... r-Debrunner theorem. In the second part we prove a (p, 2)-theorem for families with sub-quadratic union complexity in R 2 . Based on this, we introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets with sub-quadratic union complexity. Since MAX-CLIQUE for intersection graphs of fat ellipses is known to be APX-HARD and fat ellipses have sub-quadratic union complexity, this settles the approximation problem for such graphs.