On 5-Gons and 5-Holes

  • Oswin Aichholzer
  • Thomas Hackl
  • Birgit Vogtenhuber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7579)


We consider an extension of a question of Erdős on the number of k-gons in a set of n points in the plane. Relaxing the convexity restriction we obtain results on 5-gons and 5-holes (empty 5-gons). In particular, we show a direct relation between the number of non-convex 5-gons and the rectilinear crossing number, provide an improved lower bound for the number of convex 5-holes any point set must contain, and prove that the number of general 5-holes is asymptotically maximized for point sets in convex position.


discrete geometry (empty) pentagons Erdős-Szekeres type problem 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Thomas Hackl
    • 1
  • Birgit Vogtenhuber
    • 1
  1. 1.Institute for Software TechnologyUniversity of TechnologyGrazAustria

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