The chromatic number of the plane (Hadwiger–Nelson problem)

Determine how many colours are needed so that no two points of the plane at distance exactly 1 share a colour. The answer is known to be 5, 6 or 7; a concrete sub-goal is a smaller 5-chromatic unit distance graph than the 509-vertex record.

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Cite

@misc{cairn-hadwiger-nelson-problem,
  title        = {The chromatic number of the plane (Hadwiger–Nelson problem)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/hadwiger-nelson-problem}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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On the literature board

The problem

The chromatic number of the plane χ(ℝ²) is the least k such that the points of the Euclidean plane can be coloured with k colours with no two points at distance exactly 1 receiving the same colour. By the de Bruijn–Erdős theorem this equals the largest chromatic number of a finite unit distance graph.

Known status. The upper bound 7 comes from a hexagonal tiling colouring (Isbell). In 2018 de Grey exhibited a 1581-vertex unit distance graph that is not 4-colourable, proving χ(ℝ²) ≥ 5. Heule (2018) reduced this to 553 vertices using clausal proof minimisation; after the Polymath16 project, Parts (2020) obtained a 5-chromatic unit distance graph with 509 vertices and 2442 edges, the smallest known. No 6-chromatic unit distance graph is known.

What counts as progress

A 5-chromatic unit distance graph with fewer than 509 vertices (or 509 vertices and fewer edges).
A 6-chromatic unit distance graph (would raise the lower bound to 6).
Reproducible documentation of searches that fail (e.g. "no 5-chromatic subgraph of family X below n vertices"), and literature syntheses of known obstructions for 6 colours.

How it is checked — certificate format. A JSON file with (1) the vertex list, each coordinate given exactly as an element of an explicit number field (e.g. rational combinations of √3, √5, √11, ... written as symbolic expressions), (2) the edge list, and (3) a DRAT or LRAT proof that the CNF encoding "the graph is (k−1)-colourable" is unsatisfiable. A short script checks every edge has squared length exactly 1 in exact arithmetic (e.g. SymPy), regenerates the CNF deterministically from the edge list, and runs a proof checker (drat-trim or cake_lpr). Reviewers confirm vertices are distinct.

Current state

The problem