Discrete & Computational Geometry

, Volume 38, Issue 2, pp 389–397

The Empty Hexagon Theorem

Article

DOI: 10.1007/s00454-007-1343-6

Cite this article as:
Nicolas, C. Discrete Comput Geom (2007) 38: 389. doi:10.1007/s00454-007-1343-6

Abstract

Let P be a finite set of points in general position in the plane. Let C(P) be the convex hull of P and let CiP be the ith convex layer of P. A minimal convex set S of P is a convex subset of P such that every convex set of P ∩ C(S) different from S has cardinality strictly less than |S|. Our main theorem states that P contains an empty convex hexagon if C1P is minimal and C4P is not empty. Combined with the Erdos-Szekeres theorem, this result implies that every set P with sufficiently many points contains an empty convex hexagon, giving an affirmative answer to a question posed by Erdos in 1977.

Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Department of Mathematics, University of KentuckyLexington, KY 40506USA

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