Coloring the distance graphs

TL;DR

This paper explores the chromatic properties of distance graphs in Euclidean space, showing that under certain set-theoretic assumptions, the chromatic number can vary between different dimensions.

Contribution

It demonstrates the consistency, within ZF+DC, that the chromatic number of these graphs can differ between dimensions, highlighting set-theoretic influences on geometric graph properties.

Findings

Gn can have countable chromatic number in certain models

Gn+1 can have uncountable chromatic number in the same models

Set theory assumptions affect geometric graph colorings

Abstract

Let n>0 be a number. Let Gn be the graph on n-dimensional Euclidean space connecting points of rational distance. It is consistent with the choiceless theory ZF+DC that Gn has countable chromatic number yet Gn+1 does not.