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DOI: 10.4230/LIPIcs.SoCG.2018.59

Further Consequences of the Colorful Helly Hypothesis

Authors: Leonardo Martínez-Sandoval, Edgardo Roldán-Pensado, and Natan Rubin

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

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.

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Leonardo Martínez-Sandoval, Edgardo Roldán-Pensado, and Natan Rubin. Further Consequences of the Colorful Helly Hypothesis. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{martinezsandoval_et_al:LIPIcs.SoCG.2018.59,
  author =	{Mart{\'\i}nez-Sandoval, Leonardo and Rold\'{a}n-Pensado, Edgardo and Rubin, Natan},
  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 =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.59},
  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}
}

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Documents authored by Martínez-Sandoval, Leonardo

Further Consequences of the Colorful Helly Hypothesis

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