A family $$\mathcal {F}$$F of sets is said to satisfy the (p, q)-property if among any p sets of $$\mathcal {F}$$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 $$\mathbb {R}^d$$Rd that satisfies the (p, q)-property for some $$q \ge d+1$$q≥d+1, can be pierced by a fixed number (independent of the size of the family) $$f_d(p,q)$$fd(p,q) of points. The minimum such piercing number is denoted by $$\mathsf {HD} _d(p,q)$$HDd(p,q). Already in 1957, Hadwiger and Debrunner showed that whenever $$q>\frac{d-1}{d}\,p+1$$q>d-1dp+1 the piercing number is $$\mathsf {HD} _d(p,q)=p-q+1$$HDd(p,q)=p-q+1; no tight bounds on $$\mathsf {HD} _d(p,q)$$HDd(p,q) were found ever since. While for an arbitrary family of compact convex sets in $$\mathbb {R}^d$$Rd, $$d \ge 2$$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 $$\mathbb {R}^d$$Rd, 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 $$\mathsf {HD} _\mathrm{{rect}}(p,q)=p-q+1$$HDrect(p,q)=p-q+1 holds for all $$q \ge \sqrt{2p}$$q≥2p. These are the only values of q for which $$\mathsf {HD} _\mathrm{{rect}}(p,q)$$HDrect(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 $$\mathsf {HD} _{d\text {-box}}(p,q)=p-q+1$$HDd-box(p,q)=p-q+1 holds for all $$q > c' \log ^{d-1} p$$q>c′logd-1p, and in particular, $$\mathsf {HD} _\mathrm{{rect}}(p,q)=p-q+1$$HDrect(p,q)=p-q+1 holds for all $$q \ge 7 \log _2 p$$q≥7log2p (compared to $$q \ge \sqrt{2p}$$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 $$\mathcal {G}$$G of compact convex sets in $$\mathbb {R}^d$$Rd with Helly number 2 admits a (p, 2)-theorem with piercing number $$O(p^{2d-1})$$O(p2d-1), and thus, satisfies $$\mathrm {HD}_{\mathcal {G}}(p,q) = p-q+1$$HDG(p,q)=p-q+1, for a universal constant c.

中文翻译：

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从 (p, 2)-定理到紧 (p, q)-定理

如果在 $$\mathcal {F}$$F 的任何 p 个集合中，某些 q 具有非空交集，则称一个族 $$\mathcal {F}$$F 满足 (p, q)-性质. 著名的 Alon 和 Kleitman 的 (p, q)-定理断言 $$\mathbb {R}^d$$Rd 中的任何紧凑凸集族满足某些 $$q \ 的 (p, q)-性质ge d+1$$q≥d+1，可以刺穿固定数量（与家庭规模无关）$$f_d(p,q)$$fd(p,q)的点。最小的此类穿孔数由 $$\mathsf {HD} _d(p,q)$$HDd(p,q) 表示。早在 1957 年，Hadwiger 和 Debrunner 就表明，每当 $$q>\frac{d-1}{d}\,p+1$$q>d-1dp+1 时，穿孔数为 $$\mathsf {HD} _d (p,q)=p-q+1$$HDd(p,q)=p-q+1；从那以后就没有发现 $$\mathsf {HD} _d(p,q)$$HDd(p,q) 的严格界限。而对于 $$\mathbb {R}^d$$Rd 中的任意紧致凸集族，$$d \ge 2$$d≥2，a (p, 2)-property 并不意味着一个有界的穿孔数，这种界限已被证明适用于许多特定的类别。其中研究得最好的是 $$\mathbb {R}^d$$Rd 中的轴平行框类，特别是平面中的轴平行矩形。Wegner (Israel J Math 3:187–198, 1965) 和（独立地）Dol'nikov (Sibirsk Mat Ž 13(6):1272–1283, 1972) 使用 (p, 2)-theorem for axis-parallel rectangles证明 $$\mathsf {HD} _\mathrm{{rect}}(p,q)=p-q+1$$HDrect(p,q)=p-q+1 对所有 $$q \ge 成立\sqrt{2p}$$q≥2p。这些是 q 的唯一值，其中 $$\mathsf {HD} _\mathrm{{rect}}(p,q)$$HDrect(p,q) 是准确已知的。在本文中，我们提出了一种通用方法，该方法允许使用 (p, 2)-定理作为引导获得紧 (p, q)-定理，对于 Helly 数为 2 的类，即使不假设类中的集合是凸集或紧集。为了证明这种方法的优势，我们证明 $$\mathsf {HD} _{d\text {-box}}(p,q)=p-q+1$$HDd-box(p,q)= p-q+1 对所有 $$q > c' \log ^{d-1} p$$q>c'logd-1p 成立，特别是 $$\mathsf {HD} _\mathrm{{rect }}(p,q)=p-q+1$$HDrect(p,q)=p-q+1 对所有 $$q \ge 7 \log _2 p$$q≥7log2p 成立（相比 $$ q \ge \sqrt{2p}$$q≥2p，由 Wegner 和 Dol'nikov 于 40 多年前获得）。此外，对于几个类，我们提出了改进的 (p, 2)-定理，其中一些可以用作引导来获得紧 (p, q)-定理。特别是，我们证明了在 $$\mathbb {R}^d$$Rd 中的紧致凸集的任何类 $$\mathcal {G}$$G 与 Helly 数为 2 承认一个 (p, 2)-theorem with piercing数字 $$O(p^{2d-1})$$O(p2d-1)，因此，