Coloring ${\mathbb R}^n$

Abstract:

If $1 \leq m \leq n$ and $A \subseteq {\mathbb R}$, then define the graph $G(A,m,n)$ to be the graph whose vertex set is ${\mathbb R}^n$ with two vertices $x,y \in {\mathbb R}^n$ being adjacent iff there are distinct $u,v \in A^m$ such that $\|x-y\| = \|u-v\|$. For various $m$ and $n$ and various $A$, typically $A = {\mathbb Q}$ or $A = {\mathbb Z}$, the graph $G(A,m,n)$ can be properly colored with $\omega$ colors. It is shown that in some cases such a coloring $\varphi : {\mathbb R}^n \longrightarrow \omega$ can also have the additional property that if $\alpha : {\mathbb R}^m \longrightarrow {\mathbb R}^n$ is an isometric embedding, then the restriction of $\varphi$ to $\alpha (A^m)$ is a bijection onto $\omega$.