Result 158, Combinatorics

The Euclidean plane cannot be colored with five colors

Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

How many colors are needed to color every point of a flat plane so that points exactly one unit apart always differ? The manuscript reports that five cannot suffice, narrowing the answer to six or seven.

What changes?

A coloring assigns a color to every point; a color class is the set of points receiving one color. The manuscript claims that every such assignment using five colors contains two same-colored points exactly one unit apart. It imposes no regularity assumptions on those sets, so the claim is not limited to tidy regions or patterned colorings. Combined with the classical seven-color construction, this leaves six and seven as the only possible minimums.

What does that help mathematicians do?

The absence of regularity assumptions rules out escaping the obstruction by using more irregular color classes. The claim also has a direct consequence at other scales: rescaling the plane shows that five colors cannot avoid same-colored pairs at any prescribed positive distance. The restriction concerns that exact distance, not all nearby points. Whether six colors suffice remains unresolved by this result.

Are there practical applications?

The immediate value is foundational for geometric graph coloring, where points act as vertices and pairs at the forbidden distance are connected. The reported result raises the necessary number of colors for this infinite constraint system. It sharpens the target for further constructions and impossibility proofs, rather than providing a practical coloring algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The Euclidean plane is not five-colorable

September 23, 2026 62 pages Main result formalized in Lean

We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-Euclidean-plane-is-not-five-colorable-September-23-2026,
  author = {{OpenAI}},
  title = {{The Euclidean plane is not five-colorable}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf}{OAI:The-Euclidean-plane-is-not-five-colorable-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/158.md.

The Euclidean plane cannot be colored with five colors

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The Hadwiger–Nelson problem asks for the fewest colors needed to color the plane so that points at distance one have different colors. The formalized results prove that five colors do not suffice and that seven colors do suffice. The lower bound applies to arbitrary colorings, with no measurability or continuity assumption; the upper bound includes every boundary point of the coloring regions.

Comparator links

Result Comparator statement
No proper five-coloring EuclideanFiveColor.lean
Proper seven-coloring PlaneColoring.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 10 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.