The Erdös-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. Erdös and G. Szekeres showed that ES(n) exists and ES(n) ≤ + 1. About 62 years later, the upper bound has been 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..