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Updated Oct 7

Problem 508

claims/: The 3 claim pages of Problem 508, one per claimant's result; the problem's standing derives from them.

Statement. What is the chromatic number of the plane? That is, what is the smallest number of colours required to colour $\mathbb{R}^2$ such that no two points of the same colour are distance $1$ apart?

Status. Open. The site labels the problem OPEN (page last edited 22 January 2026) and records the bounds $5\le\chi\le7$ in its remarks; the claim pages record de Grey's refereed lower bound, a later result on the lower bound, and a rejected announcement of the value.

Source. erdosproblems.com/508, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #508, https://www.erdosproblems.com/508.

References.

[Cr67] H. T. Croft, Incidence incidents. Eureka 30 (1967), 22-26.
[dG18] de Grey, Aubrey D. N. J., The chromatic number of the plane is at least 5. Geombinatorics 28 (2018), no. 1, 18-31.
[OAI26] OpenAI, The Euclidean plane is not five-colorable. OpenAI Math Release preprint, 23 September 2026 (pinned PDF).

Formalization. Statement in formal-conjectures.

Current assessment

The question, as the site states it (page last edited 22 January 2026), is the value of $\chi=\chi(\mathbb R^2)$ , the least number of colors in a coloring of the plane with no two points of the same color at distance one, with no restriction on the color classes. The best bounds supported here are

6\le\chi\le7.

The lower bound is the accepted partial claim on [[problems/discrete_geometry/E0508/claims/2026_09_23_openai|OpenAI's claim page]]: Theorem 1.1 of the release preprint [OAI26] (card) proves that no coloring with five colors avoids a same-colored unit pair, for arbitrary color classes, by a transfer theorem from arbitrary colorings to measurable ones and an obstruction to five measurable labels; the same page records the seven-coloring by hexagons. Both theorems are kernel-checked in Lean, built and axiom-audited by this corpus, and the acceptance rests on that evidence alone: no outside review or refereeing of the manuscript is recorded, and the site's page does not mention it. The value of $\chi$ , six or seven, is open, so the problem is open.

Earlier bounds, as the site's remarks and the preprint's introduction record them: an equilateral triangle gives $\chi\ge3$ ; the Moser spindle and the Golomb graph give $\chi\ge4$ ; de Grey [dG18] (card; claim page) gave the first finite unit-distance graph that is not four-colorable, so $\chi\ge5$ , and later work found smaller such graphs and other proofs of that bound, among them the papers of Exoo and Ismailescu (card), Heule (card) and Parts (card); the hexagonal tiling with cells of diameter slightly less than one gives $\chi\le7$ . For the fractional chromatic number of the plane the site records the lower bound $4$ of Matolcsi, Ruzsa, Varga and Zsámboki and the upper bound $4.359\ldots$ of Croft [Cr67]. The de Bruijn–Erdős compactness theorem makes the lower bound six equivalent to the existence of a finite unit-distance graph that is not five-colorable; no such graph is exhibited, so the smallest non-five-colorable unit-distance graph is unknown.

One rejected full claim is recorded on [[problems/discrete_geometry/E0508/claims/2026_05_13_reed|Reed's claim page]]: a public manuscript first posted on 13 May 2026 announces $\chi=7$ through a circle-density argument. The preprint [OAI26] records, in a footnote, that the manuscript's proposed strict bound (angular measure less than $\pi/3$ ) on a unit-independent subset of the unit circle fails for the half-open arc $\{e^{it}:0\le t<\pi/3\}$ , which has measure $\pi/3$ and contains no unit pair, and that the manuscript's displayed formal theorem assumes that bound as a hypothesis. The site's proof-claims thread for this problem listed no claim on 6 October 2026.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (141)

Problem 188 De Grey's exclusion of four colors for the plane Reed's announcement that the plane needs seven colors OpenAI's exclusion of five colors for the plane problems/discrete_geometry/E0508/claims Problem 704 Problem 1070 Discrete and Convex Geometry Discrete and Convex Geometry discrete_geometry/aggarwal_2026_computer_aided_discovery_extremal_unit_distance Proposition 3.4: unit-distance graphs of the plane and of the sphere of radius 1/sqrt 2 differ Theorem 1.1: compactness principle for hypergraph coloring Theorem 2.4: small-tolerance separated approximate unit-distance graphs are unit-distance graphs Theorem 2.6: every finite unit-distance graph sits in a real number field Theorem 2.7: finitely many unit vectors with coordinates in (1/m)O_K Theorem 2.8: counting unit vectors with coordinates in (1/m)O_K in the plane Theorem 3.2: Raiskii lattices in every dimension discrete_geometry/ambrus_2020_density_estimates_1_avoiding_sets_via Theorem 1: a measurable planar 1-avoiding set has upper density at most 0.25442 discrete_geometry/ambrus_2023_density_planar_sets_avoiding_unit_distances Theorem 1 (p. 4): a measurable planar set avoiding unit distances has upper density at most 0.2470 discrete_geometry/axenovich_2025_ramsey_problems_graphs_euclidean_spaces_cartesian Proposition 1.7 (p. 5): chi_{C_4}(R^2) <= 4, and induced four-cycles need three colors in R^3 and, if chi(R^2) = 7, in the plane Theorem 1.1 (p. 2): forests, even cycles of length 8 or at least 12, and long odd cycles in R^n Theorem 1.5 (p. 4): H-forests, zero hypercube Turan density and Gamma_{A,B}(H,u,v) in R^n discrete_geometry/bellitto_2021_density_sets_euclidean_plane_avoiding_distance Theorem 2: planar sets avoiding distance one have density at most 0.25647 discrete_geometry/bikeev_2025_isomorphisms_unit_distance_graphs_layers Theorem 1 (p. 4): the unit distance graph of a strip determines its width Theorem 2 (p. 4): unit distance graphs of l_p layers determine their width Theorem 3 (p. 4): automorphisms of the layer R^n x [0, epsilon] are isometries discrete_geometry/cranston_2015_fractional_chromatic_number_plane Lemma 1 (p. 8): tiling the core by eight tiles with corners in a maximal independent set Theorem 2 (p. 12): the fractional chromatic number of the plane is at least 76/21 discrete_geometry/ducz_2026_unit_distance_graph_plane_independence_ratio Corollary 1 (p. 1): the fractional chromatic number of the plane exceeds 4 Corollary 2 (p. 1): the chromatic number of the plane is at least 5 discrete_geometry/engel_2025_diverse_beam_search_find_densest_known discrete_geometry/erdos_1981_applications_graph_theory_combinatorial_methods_number The chromatic number of k-space, pp. 141-142: 4 ≤ α₂ ≤ 7, Larman–Rogers and Frankl bounds, and conjecture (6) on f(n, j)Unit distance graphs in the plane, p. 142: Wormald's 4-chromatic girth-5 example, the girth conjecture, and the growth of α₂(r)discrete_geometry/exoo_2018_chromatic_number_plane_is_at_least Claim 2.1 (p. 2): 79 points force a monochromatic pair at distance sqrt(11/3)Claim 3.1 (p. 5): 49 points turn a monochromatic sqrt(11/3) pair into a monochromatic triangle Claim 4.1 (p. 6): a 627-vertex unit distance graph blocks a monochromatic 1/sqrt(3) triangle Main theorem (p. 2): the chromatic number of the plane is at least 5, a new proof discrete_geometry/exoo_2018_hadwiger_nelson_problem_two_forbidden_distances Conjecture 8.1 (p. 16): some second distance leaves a 5-chromatic plane 5-chromatic Theorem 1.4 (p. 3): the spindle method forces at least one more color Theorem 1.7 (p. 4): forbidding distances 1 and the golden ratio needs five colors

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