On piercing numbers of families satisfying the (p,q)r property

Chaya Keller, Shakhar Smorodinsky

Research output: Contribution to journalArticlepeer-review

Abstract

The Hadwiger–Debrunner number HDd(p,q) is the minimal size of a piercing set that can always be guaranteed for a family of compact convex sets in Rd that satisfies the (p,q) property. Hadwiger and Debrunner showed that HDd(p,q)≥p−q+1 for all q, and equality is attained for q>[Formula presented]p+1. Almost tight upper bounds for HDd(p,q) for a ‘sufficiently large’ q were obtained recently using an enhancement of the celebrated Alon–Kleitman theorem, but no sharp upper bounds for a general q are known. In [9], Montejano and Soberón defined a refinement of the (p,q) property: F satisfies the (p,q)r property if among any p elements of F, at least r of the q-tuples intersect. They showed that HDd(p,q)r≤p−q+1 holds for all r>(pq)−(p−d+1q−d+1); however, this is far from being tight. In this paper we present improved asymptotic upper bounds on HDd(p,q)r which hold when only a tiny portion of the q-tuples intersect. In particular, we show that for p,q sufficiently large, HDd(p,q)r≤p−q+1 holds with r=p−q/2d⋅(pq). Our bound misses the known lower bound for the same piercing number by a factor of less than pqd. Our results use Kalai's Upper Bound Theorem for convex sets, along with the Hadwiger–Debrunner theorem and the recent improved upper bound on HDd(p,q) mentioned above.

Original languageEnglish
Pages (from-to)11-18
Number of pages8
JournalComputational Geometry: Theory and Applications
Volume72
DOIs
StatePublished - 1 Jun 2018

Keywords

  • (p,q)-Theorem
  • Convexity
  • Hadwiger–Debrunner number
  • Helly-type theorems
  • Upper bound theorem

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics

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