Determine the chromatic number of the plane

Prove or refute the conjecture that the chromatic number of the planar unit-distance graph is \(\chi(\mathbb{R}^2)=6\).

Background

The paper discusses the Hadwiger–Nelson problem, for which the known bounds are 5≤χ(R2)≤75\le \chi(\mathbb{R}^2)\le 7. After presenting its transference results and possible extensions to lower dimensions, the paper explicitly conjectures that the exact planar chromatic number is six.

References

Given this new approach to improving both the upper and lower bounds for the Hadwiger-Nelson problem, we boldly conjecture the following. \begin{conjecture} $\chi(2)=6$. \end{conjecture}

— Measurable obstructions for unmeasurable colourings  (2610.12301 - Davies, 8 Oct 2026) in Introduction, final conjecture before Section \ref{sec:overview}