Colouring the Sphere

Abstract

Let $G$ be the graph with the points of the unit sphere in $\mathbb{R}^3$ as its vertices, by defining two unit vectors to be adjacent if they are orthogonal as vectors. We present a proof, based on work of Hales and Straus chromatic number of this graph is four. We also prove that the subgraph of G induced by the unit vectors with rational coordinates is 3-colourable.