Hadwiger–Nelson: is the chromatic number of the Euclidean plane 6 or 7?

The upper bound 7 comes from colouring a hexagonal tiling with hexagons of diameter slightly below 1 (Isbell, 1950). The lower bound 4 was known from 1950; the Moser spindle (W. and L. Moser, 1961) and the Golomb graph are the standard 4-chromatic witnesses. In April 2018 de Grey exhibited a 1581-vertex unit-distance graph that is not 4-colourable, proving $\chi(\mathbb R^2)\ge 5$ with SAT-solver verification; Exoo and Ismailescu gave an independent proof. Polymath16 and Heule then shrank the witness (Heule: 553 vertices, arXiv:1805.12181), and the smallest known 5-chromatic unit-distance graph has 509 vertices (Parts, Geombinatorics 29(4), 2020; the bound $v_5\le 509$ is restated by de Grey and Parts, arXiv:2303.14714). The preprint "The Euclidean plane is not five-colorable" (OpenAI, dated 23 September 2026, released in the openai/math repository in October 2026; produced by an internal OpenAI model; unrefereed) proves that every 5-colouring of the plane has a monochromatic unit-distance pair, with no regularity assumption on the colour classes, so . The proof is not a finite-graph certificate: it transfers any proper -colouring to a "weak measurable" -colouring (monochromatic unit pairs of measure zero), for every , and then excludes weak measurable 5-colourings by circle averages, Fourier decay of the circle measure and a topological argument. A finite non-5-colourable unit-distance graph therefore exists by compactness, but none is known explicitly. The Lean 4 proof (, standard axioms only) is public; Formal Conjectures marks solved with a link to it, and an independent third-party audit (13 October 2026) rebuilt it against a Mathlib-only restatement and accepted it with Comparator and the independent nanoda kernel. The same audit found that OpenAI's published Comparator configuration leaves the colouring definition unchecked; the definition itself is correct. Related variants: the fractional chromatic number of the plane is at least 4 (Matolcsi, Ruzsa, Varga, Zsámboki, arXiv:2311.10069, v4 March 2025, preprint), and Voronov proved that the plane with a small interval of forbidden distances needs at least 7 colours (arXiv:2304.10163, preprint, v3 April 2025), confirming a conjecture of Exoo. Another 2026 construction is a 2131-vertex 5-chromatic graph that avoids the Moser spindle (Haugland, arXiv:2608.04542, preprint), which is not a size record. No 6-colouring of the plane and no proof that 7 colours are needed is known; the question 6 versus 7 is open as of 2026-10-17.

None practical. The problem is a testbed for SAT-assisted combinatorial geometry and graph minimisation techniques; the methods transfer to other finite-search-with-certificate problems.

Either (a) a proof that the plane has no proper 6-colouring, giving $\chi(\mathbb R^2)=7$: for example an explicit finite unit-distance graph with a machine-checkable certificate of non-6-colourability (such as a DRAT/LRAT proof checked by a verified checker), or a formal proof; or (b) an explicit 6-colouring of the plane with no monochromatic unit-distance pair, with a proof, giving $\chi(\mathbb R^2)=6$. Either result, accepted by experts, resolves the problem. Both rely on the 2026 lower bound being upheld; if that proof failed, the question would revert to whether .