The Erdős – Szekeres Theorem : Upper Bounds and Related Results

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 ES(n) ≤ ` 2n−4 n−2 ́ + 1. 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 Erdős 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 Erdős–Szekeres theorem.

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