Finding convex sets among points in the plane

Abstract

Let g(n) denote the least value such that any g(n) points in the plane in general position contain the vertices of a convex n-gon. In 1935, Erdos and Szekeres showed that g(n) exists, and they obtained the bounds 2n-2 + 1 ≤ g(n) ≤ (2n - 4n - 2) + 1. Chung and Graham have recently improved the upper bound by 1; the first improvement since the original Erdos-Szekeres paper. We show that g(n) ≤ (2n - 4n - 2) + 7 - 2n.