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.
Current state (tracked automatically)
- Best known bounds for constant 27a: 5 ≤ C ≤ 7. Upper bound ([Had1945], [CR2017], A periodic 7-coloring obtained from a tiling of the plane by small regular hexagons (historically attributed to Isbell, 1950)); lower bound ([deG2018], First proof that ( R^2)≥ 5, via an explicit finite unit-distance graph).
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Best lower bound with an explicit value (constant 27a) | 5 [deG2018] — First proof that ( R^2)≥ 5, via an explicit finite unit-distance graph. | Tao's optimization constants |
| Best upper bound with an explicit value (constant 27a) | 7 [Had1945], [CR2017] — A periodic 7-coloring obtained from a tiling of the plane by small regular hexagons (historically attributed to Isbell, 1950). | Tao's optimization constants |
Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Tao's optimization constants (checked 2026-11-12). How it works.
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 2026-11-12}
} Also: CITATION.cff · Atom feed of results
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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.
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Current state
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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.
History of the bounds
| Year | Result | By |
|---|---|---|
| 2018 | χ ≥ 5 (1581-vertex graph) | de Grey |
| 2018 | 553-vertex 5-chromatic graph | Heule |
| 2020 | 509 vertices | Parts (Polymath16) |