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Updated Oct 7

Problem 704

claims/: The 2 claim pages of Problem 704, one per claimant's result; the problem's standing derives from them.

Statement. Let $G_n$ be the unit distance graph in $\mathbb{R}^n$ , with two vertices joined by an edge if and only if the distance between them is $1$ .

Estimate the chromatic number $\chi(G_n)$ . Does it grow exponentially in $n$ ? Does

\lim_{n\to \infty}\chi(G_n)^{1/n}

exist?

Status. Open. The site labels the problem OPEN (page last edited 10 April 2026); its proof-claims thread carried no claim as of 6 October 2026. The three questions are listed as the problem's parts: the exponential-growth question is settled by the accepted partial claims of [[problems/discrete_geometry/E0704/claims/1981_12_01_frankl_wilson|Frankl and Wilson]] and Raigorodskii, while the estimate and the existence of the limit are open, so the standing derived from the claim pages is open.

Source. erdosproblems.com/704, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #704, https://www.erdosproblems.com/704.

References.

[FrWi81] Frankl, P. and Wilson, R. M., Intersection theorems with geometric consequences. Combinatorica (1981), 357-368.
[LaRo72] Larman, D. G. and Rogers, C. A., The realization of distances within sets in Euclidean space. Mathematika (1972), 1-24.
[Pr20] Prosanov, Roman, A new proof of the Larman-Rogers upper bound for the chromatic number of the Euclidean space. Discrete Appl. Math. (2020), 115-120.
[Ra00] Raĭgorodskiĭ, A. M., On the chromatic number of a space. Uspekhi Mat. Nauk (2000), 147-148.

Formalization. None recorded.

Current assessment

The question generalizes the chromatic number of the plane, Problem 508, which is the case $n=2$ : estimate $\chi(G_n)$ for the unit distance graph of $\mathbb R^n$ , decide whether it grows exponentially in $n$ , and decide whether $\chi(G_n)^{1/n}$ converges. The site's remarks (page last edited 10 April 2026), in the corpus's words: Frankl and Wilson [FrWi81] proved exponential growth, $\chi(G_n)\ge(1+o(1))\,1.2^n$ , which answers the second question yes; Raigorodskii [Ra00] (card) raised the base to $1.239\ldots$ ; tiling by cubes gives the trivial upper bound $(2+\sqrt n)^n$ , which Larman and Rogers [LaRo72] improved to $(3+o(1))^n$ , conjecturing that the truth is $(2^{3/2}+o(1))^n$ , with $2^{3/2}\approx2.828$ ; and Prosanov [Pr20] (card) gave another proof of the Larman–Rogers bound. Whether the limit of $\chi(G_n)^{1/n}$ exists, and its value if it does, are open; the established bounds confine it to the interval $[1.239\ldots,3]$ .

Frankl and Wilson's exponential lower bound and Raigorodskii's larger base are the problem's two claim pages, accepted partial claims on their refereed publications; each proves exponential growth and so settles the exponential-growth part, and the later one sharpens the estimate. The Larman–Rogers upper bound and Prosanov's new proof of it get no claim pages: an upper bound answers none of the three questions. One recent result is recorded here because it concerns the family: OpenAI's release preprint The Euclidean plane is not five-colorable (OpenAI Math Release, 23 September 2026; card) proves that $6\le\chi(G_2)\le7$ , which is the accepted partial claim on [[problems/discrete_geometry/E0508/claims/2026_09_23_openai|Problem 508's claim page]]. It is a result about the plane alone, claims nothing about the growth of $\chi(G_n)$ in $n$ , and so gets no claim page here; its transfer theorem between arbitrary and measurable colorings is stated for the plane.

Search scope, 6 October 2026: the site's page and proof-claims thread, and the release of 23 September 2026; no further literature search.

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 (20)

Frankl and Wilson's exponential lower bound Raigorodskii's exponential lower bound with base 1.239 problems/discrete_geometry/E0704/claims Discrete and Convex Geometry Problem 703 Discrete and Convex Geometry 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)discrete_geometry/openai_2026_euclidean_plane_not_five_colorable discrete_geometry/prosanov_2020_new_proof_larman_rogers_upper_bound Theorem 1 (p. 3): the chromatic number of a normed space is bounded by its tiling parameter Section 3 result (pp. 6-7, unnumbered): the tiling parameter of the ball and the bound (3+o(1))^n discrete_geometry/raigorodskii_2000_chromatic_number_space Theorem (p. 147): the chromatic number of R^n is at least (1.239...+o(1))^n distance_problems/graham_1994_recent_trends_euclidean_ramsey_theory Section 6 (pp. 125-126): the chromatic number bounds for the plane and for n-space as reported in 1993 distance_problems/graham_2004_euclidean_ramsey_theory Problem 11.1.6 (p. 4): the chromatic number of the plane, with the chapter's definition and the bounds it reports distance_problems/graham_2010_open_problems_euclidean_ramsey_theory Unit distance survey (pp. 3-4): chromatic numbers of the plane, of 3- and 4-space and of rational space, and dense sets avoiding distance one

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