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Discrete & Computational Geometry

, Volume 19, Issue 3, pp 405–410 | Cite as

Finding Convex Sets Among Points in the Plane

  • D. Kleitman
  • L. Pachter

Abstract.

Let g(n) denote the least value such that any g(n) points in the plane in general position contain the vertices of a convex n -gon. In 1935, Erdős and Szekeres showed that g(n) exists, and they obtained the bounds \(2^{n-2}+1 \leq g(n) \leq {{2n-4} \choose {n-2}} +1. \) Chung and Graham have recently improved the upper bound by 1; the first improvement since the original Erdős—Szekeres paper. We show that \(g(n) \leq {{2n-4} \choose {n-2}}+7-2n.\) <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p405.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader>

Keywords

General Position 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag New York Inc. 1998

Authors and Affiliations

  • D. Kleitman
    • 1
  • L. Pachter
    • 1
  1. 1.Department of Mathematics, MIT, Cambridge, MA 02139, USA \{djk,lpachter\}@math.mit.eduUS

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