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Empty Convex Hexagons in Planar Point Sets

  • Published: 11 September 2007
  • Volume 39, pages 239–272, (2008)
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Empty Convex Hexagons in Planar Point Sets
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  • Tobias Gerken1 
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Abstract

Erdős asked whether every sufficiently large set of points in general position in the plane contains six points that form a convex hexagon without any points from the set in its interior. Such a configuration is called an empty convex hexagon. In this paper, we answer the question in the affirmative. We show that every set that contains the vertex set of a convex 9-gon also contains an empty convex hexagon.

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Authors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85747, Garching, Germany

    Tobias Gerken

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  1. Tobias Gerken
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Correspondence to Tobias Gerken.

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Gerken, T. Empty Convex Hexagons in Planar Point Sets. Discrete Comput Geom 39, 239–272 (2008). https://doi.org/10.1007/s00454-007-9018-x

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  • Received: 13 July 2005

  • Revised: 04 September 2006

  • Published: 11 September 2007

  • Issue date: March 2008

  • DOI: https://doi.org/10.1007/s00454-007-9018-x

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Keywords

  • Erdős-Szekeres problem
  • Ramsey theory
  • Convex polygons and polyhedra
  • Empty hexagon problem

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