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

Abstract

A family $\mathcal {F}$ of sets is said to satisfy the (p, q)-property if among any p sets of $\mathcal {F}$ 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 $\mathbb {R}^d$ that satisfies the (p, q)-property for some $q \ge d+1$, 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 $\mathsf {HD} _d(p,q)$. Already in 1957, Hadwiger and Debrunner showed that whenever $q>\frac{d-1}{d}\,p+1$ the piercing number is $\mathsf {HD} _d(p,q)=p-q+1$; no tight bounds on $\mathsf {HD} _d(p,q)$ were found ever since. While for an arbitrary family of compact convex sets in $\mathbb {R}^d$, $d \ge 2$, 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 $\mathbb {R}^d$, 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 $\mathsf {HD} _\mathrm{{rect}}(p,q)=p-q+1$ holds for all $q \ge \sqrt{2p}$. These are the only values of q for which $\mathsf {HD} _\mathrm{{rect}}(p,q)$ 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 $\mathsf {HD} _{d\text {-box}}(p,q)=p-q+1$ holds for all $q > c' \log ^{d-1} p$, and in particular, $\mathsf {HD} _\mathrm{{rect}}(p,q)=p-q+1$ holds for all $q \ge 7 \log _2 p$ (compared to $q \ge \sqrt{2p}$, 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 $\mathcal {G}$ of compact convex sets in $\mathbb {R}^d$ with Helly number 2 admits a (p, 2)-theorem with piercing number $O(p^{2d-1})$, and thus, satisfies $\mathrm {HD}_{\mathcal {G}}(p,q) = p-q+1$, for a universal constant c.