Colouring the Triangles Determined by a Point Set

TL;DR

This paper investigates the chromatic number of intersection graphs formed by open triangles determined by a point set in the plane, providing new bounds that improve understanding of geometric graph coloring.

Contribution

It establishes an upper bound of approximately n^3/19.259 for the chromatic number for arbitrary point sets, refining previous bounds.

Findings

Chromatic number is at least n^3/27+O(n^2)

Chromatic number is at most n^3/19.259+O(n^2) for arbitrary sets

Exact bounds depend on the point set configuration

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 n^3/27+O(n^2), and if P is in convex position, the answer is n^3/24+O(n^2). We prove that for arbitrary P, the chromatic number is at most n^3/19.259+O(n^2).