Colouring the triangles determined by a point set

Abstract

Let P be a set of n points in general position in the plane. We study the chromatic number of the intersection graph of the open triangles determined by P. It is known that this chromatic number is at least n³/27+O(n²) and, if P is in convex position, the answer is n³/24+O(n²). We prove that for arbitrary P, the chromatic number is at most n³/19.259+O(n²).