Algebraic formulation of Hadwiger-Nelson problem

This is a part of Polymath16 - for the main page, see Hadwiger-Nelson problem.

Given a unit distance graph [math]\displaystyle{ G }[/math], the edges of the graph are formed by a set of vectors [math]\displaystyle{ V }[/math] of unit length. Often a single vector is used many times in the same graph for different edges. The vectors generate an infinite group [math]\displaystyle{ \mathcal{G} }[/math] under addition. The Cayley graph [math]\displaystyle{ \Gamma(\mathcal{G},S) }[/math] is a unit distance graph that will contain [math]\displaystyle{ G }[/math] as a subgraph. A colouring of the Cayley graph reduces to a colouring of [math]\displaystyle{ G }[/math], and is an efficient way to set an upper bound on its chromatic number.