Consider the graph on $\mathbb{Q}\times\mathbb{Q}$ where two members of $\mathbb{Q}\times\mathbb{Q}$ form an edge if and only if their distance is $1$. Is that graph bipartite? If not, what is its chromatic number? (It is [known](https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem) to be $\leq 7$.)



Is the Hadwiger-Nelson graph restricted to $\mathbb{Q}\times\mathbb{Q}$ bipartite?
Yes, the chromatic number is $2$. This result is due to Douglas R. Woodall, “Distances realized by sets covering the plane”, *J. Combinatorial Theory* **14** (1973), 187–200. See also Alexander Soifer, *The Mathematical Coloring Book (Mathematics of Coloring and the Colorful Life of Its Creators)*, Springer (2009), §11.2. See also my unpublished¹ note [“The Hadwiger-Nelson problem over certain fields”](https://arxiv.org/abs/1509.07023), esp. ¶1.6.


1. Because it turned out that (unbeknownst to me) most of the results contained therein were already obtained earlier in an (also unpublished) note by Eric Moorhouse, [“On the Chromatic Numbers of Planes”](https://ericmoorhouse.org/pub/chromatic.pdf), which is also relevant here.


