The Erdos-Szekeres problem on points in convex position – a survey

Abstract

In 1935 Erdős and Szekeres proved that for any integer n ≥ 3 n \ge 3 there exists a smallest positive integer N ( n ) N(n) such that any set of at least N ( n ) N(n) points in general position in the plane contains n n points that are the vertices of a convex n n -gon. They also posed the problem to determine the value of N ( n ) N(n) and conjectured that N ( n ) = 2 n − 2 + 1 N(n) = 2^{n-2} +1 for all n ≥ 3. n \ge 3. Despite the efforts of many mathematicians, the Erdős-Szekeres problem is still far from being solved. This paper surveys the known results and questions related to the Erdős-Szekeres problem in the plane and higher dimensions, as well as its generalizations for the cases of families of convex bodies and the abstract convexity setting.