Finding Convex Sets Among Points in the Plane

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, Erdős and Szekeres showed that g(n) exists, and they obtained the bounds 2^(n−2) + 1 ≤ g(n) ≤ (^(2n−4)_(n−2)) + 1. Chung and Graham have recently improved the upper bound by 1; the first improvement since the original Erdős—Szekeres paper. We show that g(n) ≤ (^(2n−4)_(n−2)) + 7 − 2n.