Coloring the distance graph in three dimensions

TL;DR

This paper explores the chromatic properties of distance graphs in three and four dimensions, revealing that their chromatic numbers can differ dramatically under certain set-theoretic assumptions.

Contribution

It demonstrates, under specific axioms, that the three-dimensional distance graph can be countably chromatic while the four-dimensional one can be uncountably chromatic, highlighting dimension-dependent complexity.

Findings

G3 has countable chromatic number under certain set-theoretic assumptions.

G4 can have uncountable chromatic number under the same assumptions.

The results depend on the consistency relative to an inaccessible cardinal.

Abstract

Let Gn be the graph on n-dimensional Euclidean space connecting points of rational Euclidean distance. It is consistent relative to an inaccessible cardinal that ZF+DC holds and G3 has countable chromatic number, yet G4 has uncountable chromatic number.