Establish the three-dimensional quadratic-extension equality

Establish whether there exists a positive integer \(n\) such that the chromatic number of the three-dimensional unit-distance graph equals the chromatic number of the graph on \((\sqrt n\,\mathbb{Z})^3\), namely \(\chi(\mathbb{R}^3)=\chi((\sqrt n\,\mathbb{Z})^3)\).

Background

The main transference theorem shows that in four dimensions there exists a positive integer nn for which the ordinary chromatic number of the four-dimensional Euclidean unit-distance graph agrees with that of the corresponding quadratic-extension lattice graph. The authors state that they conjecture an analogous equality in three dimensions. Such an equality would reduce the three-dimensional problem to an arithmetic graph-colouring problem.

References

For instance, in the four dimensional case the proof of Theorem \ref{thm:main} shows that there exists some positive integer $n$ such that $\chi(4)=\chi((\sqrt n)4)$. We conjecture that this also holds in the three dimensional case.

— Measurable obstructions for unmeasurable colourings  (2610.12301 - Davies, 8 Oct 2026) in Introduction, paragraph following Theorem \ref{thm:main} and preceding the conjecture \(\chi(\mathbb{R}^2)=6\)