From a (p, 2)-Theorem to a Tight (p, q)-Theorem

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 of 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-1dp+1 the piercing number is HD_d(p, q) = p- q+ 1 ; no tight bounds on 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 classes. The best-studied among them is the class of axis-parallel boxes in R^d, and specifically, axis-parallel rectangles in the plane. Wegner (Israel J Math 3:187–198, 1965) and (independently) Dol’nikov (Sibirsk Mat Ž 13(6):1272–1283, 1972) used a (p, 2)-theorem for axis-parallel rectangles to show that HD_rect(p, q) = p- q+ 1 holds for all q≥2p. These are the only values of q for which HD_rect(p, q) is known exactly. In this paper we present a general method which allows using a (p, 2)-theorem as a bootstrapping to obtain a tight (p, q)-theorem, for classes with Helly number 2, even without assuming that the sets in the class are convex or compact. To demonstrate the strength of this method, we show that HD_d-box(p, q) = p- q+ 1 holds for all q> c^′log ^d-1p, and in particular, HD_rect(p, q) = p- q+ 1 holds for all q≥ 7 log ₂p (compared to q≥2p, obtained by Wegner and Dol’nikov more than 40 years ago). In addition, for several classes, we present improved (p, 2)-theorems, some of which can be used as a bootstrapping to obtain tight (p, q)-theorems. In particular, we show that any class G of compact convex sets in R^d with Helly number 2 admits a (p, 2)-theorem with piercing number O(p^2d-1) , and thus, satisfies HD _G(p, q) = p- q+ 1 , for a universal constant c.