Forced convex n-gons in the plane

Abstract

In a seminal paper from 1935. Erdos and Szekeres showed that for each n there exists aleast value g(n) such that any subset of g(n) points in the plane in general position must always contain the vertices of a convex n-gon. In particular, they obtained the bounds 2n-2 + 1 ≤ g(n) ≤ (2n - 4n - 2) + 1, which have stood unchanged since then. In this paper we remove the +1 from the upper bound for n ≥ 4.