Embedding Euclidean Distance Graphs in $\mathbb{R}^n$ and $\mathbb{Q}^n$

Abstract

For $S \subseteq \mathbb{R}$, positive integer $n$, and $d > 0$, let $G(S^n, d)$ be the graph whose vertex set is $S^n$ where any two vertices are adjacent if and only if they are Euclidean distance $d$ apart. The primary question we will consider in our work is as follows. Given $n$ and distance $d$ actually realized as a distance between points of the rational space $\mathbb{Q}^n$, does there exist a finite graph $G$ that appears as a subgraph of $G(\mathbb{Q}^n, d)$ but not as a subgraph of $G(\mathbb{R}^{n-1}, 1)$? We answer this question affirmatively for $n \leq 5$, and along the way, resolve a few related questions as well.