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DOI: 10.4230/LIPIcs.SoCG.2018.59
URN: urn:nbn:de:0030-drops-87726
URL: http://drops.dagstuhl.de/opus/volltexte/2018/8772/
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Martínez-Sandoval, Leonardo ; Roldán-Pensado, Edgardo ; Rubin, Natan

Further Consequences of the Colorful Helly Hypothesis

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Abstract

Let F be a family of convex sets in R^d, which are colored with d+1 colors. We say that F satisfies the Colorful Helly Property if every rainbow selection of d+1 sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family F there is a color class F_i subset F, for 1 <= i <= d+1, whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension d >= 2 there exist numbers f(d) and g(d) with the following property: either one can find an additional color class whose sets can be pierced by f(d) points, or all the sets in F can be crossed by g(d) lines.

BibTeX - Entry

@InProceedings{martnezsandoval_et_al:LIPIcs:2018:8772,
  author =	{Leonardo Mart{\'i}nez-Sandoval and Edgardo Rold{\'a}n-Pensado and Natan Rubin},
  title =	{{Further Consequences of the Colorful Helly Hypothesis}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{59:1--59:14},
  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/2018/8772},
  URN =		{urn:nbn:de:0030-drops-87726},
  doi =		{10.4230/LIPIcs.SoCG.2018.59},
  annote =	{Keywords: geometric transversals, convex sets, colorful Helly-type theorems, line transversals, weak epsilon-nets, transversal numbers}
}

Keywords: geometric transversals, convex sets, colorful Helly-type theorems, line transversals, weak epsilon-nets, transversal numbers
Seminar: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 24.05.2018


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