Problem 704

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?

Source: erdosproblems.com/704

Graph theory
Geometry
Chromatic number

Wiki page

Status

Open

No claim settles this problem.

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 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.

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.

Problem 704

Status

Proof claims

Discussion