A family F of convex sets in ℝ^d satisfies the (p,q)-property if among any p members, some q have non-empty intersection. Question: find the minimum piercing number \(HD_{d}\)(p,q) = the smallest number of points needed to stab every member. Alon–Kleitman proved \(HD_{d}\)(p,q) < ∞; exact values unknown for most (p,q,d).
Hadwiger and Debrunner formulated the framework in 1957. The (p,2) case relates to Helly's theorem (\(HD_{d}\)(d+1,d+1)=1). Alon and Kleitman's 1992 '(p,q)-theorem' proved boundedness generally — a major result recognised with the Gödel Prize. The sharp bounds were found by Karasev for limited cases; the general exact answer remains one of discrete geometry's most-wanted numbers.
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