A family F of sets is said to satisfy the (p,q)-property if among any p sets of F some q have a non-empty intersection. The celebrated (p,q)-theorem of Alon and Kleitman asserts that any family of compact convex sets in R^d that satisfies the (p,q)-property for some q>=d+1, can be pierced by a fixed number (independent on the size of the family) f_d(p,q) of points. The minimum such piercing number is denoted by HD_d(p,q). Already in 1957, Hadwiger and Debrunner showed that whenever q>(d-1)p/d, the piercing number is HD_d(p,q)=p-q+1; no exact values of HD_d(p,q) were found ever since.
While for an arbitrary family of compact convex sets in R^d, d>=2, a (p,2)-property does not imply a bounded piercing number, such bounds were proved for numerous specific families. The best-studied among them is axis-parallel boxes in R^d, and specifically, axis-parallel rectangles in the plane. Wegner and (independently) Dol'nikov used a (p,2)-theorem for axis-parallel rectangles to show that HD_{rect}(p,q)=p-q+1 holds for all q>sqrt{2p}. These were the only values of q for which HD_{rect}(p,q) was known exactly.
In this talk we present a general method which allows using a (p,2)-theorem as a bootstrapping to obtain a tight (p,q)-theorem, for families with Helly number 2, even without assuming that the sets in the family are convex or compact. To demonstrate the strength of this method, we obtain an improvement of Wegner and Dol'nikov's result. Namely, we show that HD_{d-box}(p,q)=p-q+1 holds for all q>c' log^{d-1} p, and in particular, HD_{rect}(p,q)=p-q+1 holds for all q >= 7log p (compared to q>=sqrt{2p} of Wegner and Dol'nikov).
Joint work with Shakhar Smorodinsky