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Further Consequences of the Colorful Helly Hypothesis

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

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Leonardo Martínez-Sandoval
Edgardo Roldán-Pensado
Natan Rubin

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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)
https://doi.org/10.4230/LIPIcs.SoCG.2018.59

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.
Keywords
  • geometric transversals
  • convex sets
  • colorful Helly-type theorems
  • line transversals
  • weak epsilon-nets
  • transversal numbers

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