# P52: Hadwiger-Nelson problem

- ID: `P52`
- Reference: `hadwiger-nelson-problem`
- Page: https://theoremdb.org/statements/P52
- Record maturity: Reviewed problem with recorded work

## Problem

Determine the chromatic number \(\chi(\mathbb{R}^2)\) of the unit-distance graph on the Euclidean plane, whose vertices are points of \(\mathbb{R}^2\) and whose edges join pairs at distance \(1\).

### Context

Finite unit-distance graphs give lower bounds, while explicit colorings of the whole plane give upper bounds.

### Problem setup

- **Definition (The chromatic number of the plane).** The chromatic number of the plane is the chromatic number of the graph whose vertices are all planar points and whose edges join points exactly one unit apart.
- **Definition (A valid coloring may assign colors without any measurability requirement).** A valid coloring may assign colors without any measurability requirement.
- **Remark.** Finite unit-distance graphs give lower bounds, while explicit colorings of the whole plane give upper bounds.

### What counts as a solution

- Determine the exact value by giving a coloring with k colors and proving that every coloring with fewer than k colors creates a monochromatic unit-distance pair.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: De Grey proves the lower bound 5 by a finite unit-distance graph, while the classical hexagonal construction gives the upper bound 7. The current unrestricted value is 5, 6, or 7. Exact unresolved remainder: Decide whether the chromatic number of the Euclidean plane's unit-distance graph is 5, 6, or 7, with a coloring for the upper bound and a finite or otherwise rigorous obstruction for the lower bound. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: De Grey proves the lower bound 5 by a finite unit-distance graph, while the classical hexagonal construction gives the upper bound 7. The current unrestricted value is 5, 6, or 7. Exact unresolved remainder: Decide whether the chromatic number of the Euclidean plane's unit-distance graph is 5, 6, or 7, with a coloring for the upper bound and a finite or otherwise rigorous obstruction for the lower bound.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: De Grey proves the lower bound 5 by a finite unit-distance graph, while the classical hexagonal construction gives the upper bound 7. The current unrestricted value is 5, 6, or 7.

Exact unresolved remainder: Decide whether the chromatic number of the Euclidean plane's unit-distance graph is 5, 6, or 7, with a coloring for the upper bound and a finite or otherwise rigorous obstruction for the lower bound.

### Background and intake notes

- Original intake status: The cited paper proves the lower bound 5, while the standard upper bound is 7; the exact value remains one of 5, 6, or 7. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The lower bound, upper bound, and exact open status were checked on 2026-07-22.
- Variants imposing measurable color classes or forbidding an interval of distances are different problems.

- Recorded example: A regular hexagonal tiling construction gives a finite upper bound, while finite unit-distance graphs force at least five colors.

### Open directions

- **Route 1** (reported): Determine the exact value by giving a coloring with k colors and proving that every coloring with fewer than k colors creates a monochromatic unit-distance pair. [1](#reference-1)

### Computational notes

- Computer searches can discover finite obstruction graphs and candidate colorings, but a whole-plane upper bound needs a mathematical construction.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `hadwiger-nelson-problem`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Aubrey D. N. J. de Grey, “The chromatic number of the plane is at least 5”. arXiv:1804.02385 (2018). Aubrey de Grey, arXiv:1804.02385, abstract and construction https://arxiv.org/abs/1804.02385
   - Also cited at abstract and finite unit-distance graph construction
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:1804.02385, checked 2026-07-31; checked 2026-07-31
   - Source use: original_summary
   - The cited paper proves the lower bound 5, while the standard upper bound is 7; the exact value remains one of 5, 6, or 7. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - Proves the unrestricted lower bound 5.
   - Source named by the research packet.
2. <a id="reference-2"></a>Georgy Sokolov and Vsevolod Voronov, “On the chromatic number of the plane for map-type colorings”. arXiv:2502.01958 (2025). abstract and hypotheses restricting color classes to map-type or polygonal regions https://arxiv.org/abs/2502.01958
   - preprint; primary source; arXiv:2502.01958v1; checked 2026-08-01
   - Source use: original_summary
   - Proves a seven-color lower bound only for a restricted coloring class and does not settle arbitrary colorings of the plane.