Let $$n\geqslant 1$$ be a number. $${\Gamma }_n$$ the graph on $$\mathbb {R}^n$$ connecting points of rational Euclidean distance. It is consistent with choiceless set theory $$\textrm{ZF}\,{+}\,\textrm{DC}$$ that has countable chromatic number, yet number $$\Gamma _{n+1}$$ uncountable.