Axiom of choice and chromatic number of \(R^n\)
The chromatic number of the following graph is considered. The vertices are the points of the Euclidean space \({\mathbb R}^n\), the points \({\mathbf x}\) and \({\mathbf y}\) are joined if \({\mathbf x}-{\mathbf y}-\sqrt{2}{\mathbf e}_i\in {\mathbb Q}^n\) holds for some \(1\leq i\leq n\) (here \({\mathbf e}_1,\dots,{\mathbf e}_n\) is the standard basis). Assuming AC, the axiom of choice, the chromatic number is 2, it is, however, uncountable if \(\text{ AC}_{\aleph_0}\) holds and every set of reals is measurable.
scientific article; zbMATH DE number 2164803