Geometric variants of Hall's Theorem through Sperner's Lemma

Leonardo Martinez, Luis Montejano

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Hall's marriage theorem can be regarded as a condition for the existence of a rainbow set of distinct points. Motivated by this interpretation, we introduce the notion of geometric Hall-type problems. We prove that a linear Hall-type condition guarantees the existence of a rainbow set of pairwise disjoint unit balls in Rn. We also prove that a quadratic Hall-type condition guarantees the existence of a rainbow set of points in general position on the plane. To prove these results, we present and extend a topological technique by Aharoni and Haxell that uses Sperner's lemma in tight triangulations of the k-dimensional simplex.

Original languageEnglish
Pages (from-to)127-132
Number of pages6
JournalElectronic Notes in Discrete Mathematics
Volume44
DOIs
StatePublished - 5 Nov 2013
Externally publishedYes

Keywords

  • Hall's theorem
  • Kissing number
  • Sperner's lemma
  • Topological proof

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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