The Erdos-Szekeres problem on points in convex position - A survey

In 1935 Erdos and Szekeres proved that for any integer n greater than or eq ual to 3 there exists a smallest positive integer N(n) such that any set of at least N(n) points in general position in the plane contains n points th at are the vertices of a convex n-gon. They also posed the problem to deter mine the value of N(n) and conjectured that N(n) = 2(n-2) + 1 for all n gre ater than or equal to 3. Despite the efforts of many mathematicians, the Erdos-Szekeres problem is s till far from being solved. This paper surveys the known results and questi ons related to the Erdos-Szekeres problem in the plane and higher dimension s, as well as its generalizations for the cases of families of convex bodie s and the abstract convexity setting.