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

Chaya Keller, Shakhar Smorodinsky

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

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 Rd 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) fd(p, q) of points. The minimum such piercing number is denoted by HDd(p, q). Already in 1957, Hadwiger and Debrunner showed that whenever q>d-1dp+1 the piercing number is HDd(p, q) = p- q+ 1 ; no tight bounds on HDd(p, q) were found ever since. While for an arbitrary family of compact convex sets in Rd, 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 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 HDrect(p, q) = p- q+ 1 holds for all q≥2p. These are the only values of q for which 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 HDd-box(p, q) = p- q+ 1 holds for all q> clog d-1p, and in particular, HDrect(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 Rd with Helly number 2 admits a (p, 2)-theorem with piercing number O(p2d-1) , and thus, satisfies HD G(p, q) = p- q+ 1 , for a universal constant c.

Original languageEnglish
Pages (from-to)821-847
Number of pages27
JournalDiscrete and Computational Geometry
Volume63
Issue number4
DOIs
StatePublished - 1 Jun 2020

Keywords

  • (p,q)-Theorem
  • Axis-parallel rectangles
  • Convexity
  • Hadwiger–Debrunner numbers
  • Helly-type theorems

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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