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DOI: 10.4230/LIPIcs.SoCG.2018.51
URN: urn:nbn:de:0030-drops-87640
URL: http://drops.dagstuhl.de/opus/volltexte/2018/8764/
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Keller, Chaya ; Smorodinsky, Shakhar

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

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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 R^d that satisfies the (p,q)-property for some q >= d+1, can be pierced by a fixed number (independent on 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-1)/d p+1 the piercing number is {HD}_d(p,q)=p-q+1; no exact values of {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 families. The best-studied among them is axis-parallel boxes in R^d, and specifically, axis-parallel rectangles in the plane. Wegner (1965) and (independently) Dol'nikov (1972) used a (p,2)-theorem for axis-parallel rectangles to show that {HD}_{rect}(p,q)=p-q+1 holds for all q>sqrt{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 families with Helly number 2, even without assuming that the sets in the family 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-1} p, and in particular, {HD}_{rect}(p,q)=p-q+1 holds for all q >= 7 log_2 p (compared to q >= sqrt{2p}, obtained by Wegner and Dol'nikov more than 40 years ago). In addition, for several classes of families, 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 family F 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}_{F}(p,q)=p-q+1 for all q>cp^{1-1/(2d-1)}, for a universal constant c.

BibTeX - Entry

@InProceedings{keller_et_al:LIPIcs:2018:8764,
  author =	{Chaya Keller and Shakhar Smorodinsky},
  title =	{{From a (p,2)-Theorem to a Tight (p,q)-Theorem}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{51:1--51:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Bettina Speckmann and Csaba D. T{\'o}th},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8764},
  URN =		{urn:nbn:de:0030-drops-87640},
  doi =		{10.4230/LIPIcs.SoCG.2018.51},
  annote =	{Keywords: (p,q)-Theorem, convexity, transversals, (p,2)-theorem, axis-parallel rectangles}
}

Keywords: (p,q)-Theorem, convexity, transversals, (p,2)-theorem, axis-parallel rectangles
Seminar: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 24.05.2018


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