The Erdös—Szekeres Theorem: Upper Bounds and Related Results

Summary

Let ES(n) denote the least integer such that among any ES(n) points in general position in the plane there are always n in convex position. In 1935, P. Erdős and G. Szekeres showed that ES(n) exists and Six decades later, the upper bound was slightly improved by Chung and Graham, a few months later it was further improved by Kleitman and Pachter, and another few months later it was further improved by the present authors. Here we review the original proof of Erdös and Szekeres, the improvements, and finally we combine the methods of the first and third improvements to obtain yet another tiny improvement. We also briefly review some of the numerous results and problems related to the Erdös-Szekeres theorem.

1. Introduction

In 1933, Esther Klein raised the following question. Is it true that for every n there is a least number — which we will denote by ES(n) — such that among any ES(n) points in general position in the plane there are always n in convex position? This question was answered in the affirmative in a classical paper of Erdős and Szekeres [1935]. In fact, they showed (see also [Erdős and Szekeres 1960/1961]) that

The lower bound, 2^n-2 +1, is sharp for ⁿ = 2, 3, 4, 5 and has been conjectured to be sharp for all n. However, the upper bound, was not improved for 60 years. Recently, Chung and Graham [1998] managed to improve it by 1.