The *Hadwiger-Nelson graph* on $\mathbb{R}^n$ is defined to be $(\mathbb{R}^n,E\_n)$ where $$E\_n = \big\{\{x,y\}: x,y\in \mathbb{R}^n \text{ and } |x-y|=1\big\},$$ where $|\cdot|$ denotes the Euclidean distance in $\mathbb{R}^n$.


For $n>1$, is there a [matching](https://en.wikipedia.org/wiki/Matching_(graph_theory)) $M\subseteq E\_n$ such that $\bigcup M = \mathbb{R}^n$?



Does the Hadwiger-Nelson graph have a perfect matching?
You can find such a perfect matching on $\mathbb R$, by taking the pairs $\{x,x+1\}$ whenever $\lfloor x\rfloor$ is even. Next, you can put a copy of this matching on every parallel translate of some line in $\mathbb R^n$ to get a perfect matching in $E\_n$.


