On the Erdős-Szekeres convex polygon problem

Abstract

Let E S ( n ) ES(n) be the smallest integer such that any set of E S ( n ) ES(n) points in the plane in general position contains n n points in convex position. In their seminal 1935 paper, Erdős and Szekeres showed that E S ( n ) ≤ ( 2 n − 4 n − 2 ) + 1 = 4 n − o ( n ) ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)} . In 1960, they showed that E S ( n ) ≥ 2 n − 2 + 1 ES(n) \geq 2^{n-2} + 1 and conjectured this to be optimal. In this paper, we nearly settle the Erdős-Szekeres conjecture by showing that E S ( n ) = 2 n + o ( n ) ES(n) =2^{n +o(n)} .