Abstract
Let \(n\geqslant 1\) be a number. Let \({\Gamma }_n\) be the graph on \(\mathbb {R}^n\) connecting points of rational Euclidean distance. It is consistent with choiceless set theory \(\textrm{ZF}\,{+}\,\textrm{DC}\) that \({\Gamma }_n\) has countable chromatic number, yet the chromatic number of \(\Gamma _{n+1}\) is uncountable.
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Zapletal, J. Coloring the distance graphs. European Journal of Mathematics 9, 66 (2023). https://doi.org/10.1007/s40879-023-00665-6
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DOI: https://doi.org/10.1007/s40879-023-00665-6