A modular version of the Erdős– Szekeres theorem

Abstract

Bialostocki, Dierker, and Voxman proved that for any n = p +2, there is an integer B(n; p) with the following property. Every set of B(n; p) points in general position in the plane has n points in convex position such that the number of points in the interior of their convex hull is 0 mod p. They conjectured that the same is true for all pairs n = 3, p =2. In this note, we show that every su&ciently large point set determining no triangle with more than one point in its interior has n elements that form the vertex set of an empty convex n-gon. As a consequence, we show that the above conjecture is true for all n =5p=6+O(1).