Further Consequences of the Colorful Helly Hypothesis

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

Research output: Contribution to journalArticlepeer-review


Let F be a family of convex sets in Rd, 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 Fi⊂ 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.

Original languageAmerican English
Pages (from-to)848-866
Number of pages19
JournalDiscrete and Computational Geometry
Issue number4
StatePublished - 1 Jun 2020


  • Colorful Helly-type theorems
  • Convex sets
  • Geometric transversals
  • Line transversals
  • Transversal numbers
  • Weak epsilon-nets

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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