Note on the Erdos - Szekeres Theorem

Let g(n) denote the least integer such that among any g(n) points in general position in the plane there are always n in convex position. In 1935, P. Erdős and G. Szekeres showed that g(n) exists and \(2^{n-2}+1\le g(n)\le {2n-4\choose n-2}+1\) . Recently, the upper bound has been slightly improved by Chung and Graham and by Kleitman and Pachter. In this paper we further improve the upper bound to \(g(n)\le {2n-5\choose n-2}+2.\) <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>19n3p457.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader>

Authors and Affiliations

1. Courant Institute, NYU, 251 Mercer Street, New York, NY 10012, USA toth@cims.nyu.edu , , , , , , US

   G. Tóth

2. DIMACS Center, Rutgers University, Piscataway, NJ 08855, USA , , , , , , US

   P. Valtr

Received July 1, 1997, and in revised form July 14, 1997.

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