A Triangle-free, 4-chromatic $\mathbb{Q}^3$ Euclidean Distance Graph Scavenger Hunt!

Abstract

For $d > 0$, define $G(\mathbb{Q}^3, d)$ to be the graph whose set of vertices is the rational space $\mathbb{Q}^3$, where two vertices are adjacent if and only if they are a Euclidean distance $d$ apart. Let $\chi(\mathbb{Q}^3, d)$ be the chromatic number of such a graph or, in other words, the minimum number of colors needed to color the points of $\mathbb{Q}^3$ so that no two points at distance $d$ apart receive the same color. An open problem, originally posed by Benda and Perles in the 1970s, asks if there exists $d$ such that $\chi(\mathbb{Q}^3, d) = 3$. Through numerous efforts over the years, $\chi(\mathbb{Q}^3, d)$ has been determined for many values of $d$, and for all those distances $d$ where $\chi(\mathbb{Q}^3, d)$ has not been exactly pinned down, it is known that $\chi(\mathbb{Q}^3, d) \in \{3,4\}$. In our work, we detail several search algorithms we have employed to find $4$-chromatic subgraphs of various graphs $G(\mathbb{Q}^3, d)$ whose chromatic number was previously unknown. Ultimately, we conjecture that no $3$-chromatic $G(\mathbb{Q}^3, d)$ exists. Along the way, we pose a few related questions that we feel are of interest in their own right.