[#P52] Hadwiger-Nelson problem

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 \(1\).

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Definitions and notation

1Context

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

2Problem setup

Definition 1 (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 2 (A valid coloring may assign colors without any measurability requirement). A valid coloring may assign colors without any measurability requirement.

Remark 1. Finite unit-distance graphs give lower bounds, while explicit colorings of the whole plane give upper bounds.

3What 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.

1Status

What counts as a solution

Current status (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.^[1][2]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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][2] Relevance to this problem For Hadwiger-Nelson problem, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details 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. Open full record →
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.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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] Relevance to this problem For Hadwiger-Nelson problem, record hadwiger-nelson-problem-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

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 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Variants imposing measurable color classes or forbidding an interval of distances are different problems.

Recorded example 1. A regular hexagonal tiling construction gives a finite upper bound, while finite unit-distance graphs force at least five colors.

Computational notes

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

2See also

Grid points with no three collinear and no four concyclicdiscrete geometry
Largest equilateral-triangle-free subset of the fifteen-row triangular latticediscrete geometry
Most lattice points with all pairwise slopes distinct in a ten by ten griddiscrete geometry

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Plain text

“Hadwiger-Nelson problem.” TheoremDB. P52. Problem statement; statement text SHA-256 c2231dc29708895b65e0a6896bb75154fa323087095ca4f509b180b954ac621c. https://theoremdb.org/statement/?ref=P52

BibTeX

@misc{theoremdb-problem-c2231dc29708895b65e0a6896bb75154fa323087095ca4f509b180b954ac621c,
  title = {{Hadwiger-Nelson problem}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 c2231dc29708895b65e0a6896bb75154fa323087095ca4f509b180b954ac621c},
  url = {https://theoremdb.org/statement/?ref=P52}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. 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. ↗preprint · primary source · arXiv:1804.02385, checked 2026-07-31 · checked 2026-07-31Source 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.Also cited at abstract and finite unit-distance graph construction.Also cited at Editorial research route recorded 2026-07-31.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.
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. ↗preprint · primary source · arXiv:2502.01958v1 · checked 2026-08-01Source use: original summary.Proves a seven-color lower bound only for a restricted coloring class and does not settle arbitrary colorings of the plane.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P52] Hadwiger-Nelson problem

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

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Computational notes

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Computational notes

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Background

Definitions

What counts as a solution

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