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Erdős #704 · graph theory / geometry

Let GnG_n be the unit distance graph in Rn\mathbb{R}^n, with two vertices joined by an edge if and only if the distance between them is 11.

Estimate the chromatic number χ(Gn)\chi(G_n). Does it grow exponentially in nn? …

MSC 05C · Graph theory

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Problem statement

Let GnG_n be the unit distance graph in Rn\mathbb{R}^n, with two vertices joined by an edge if and only if the distance between them is 11.

Estimate the chromatic number χ(Gn)\chi(G_n). Does it grow exponentially in nn? Does

lim⁡n→∞χ(Gn)1/n\lim_{n\to \infty}\chi(G_n)^{1/n}

exist?

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Background

A generalisation of the Hadwiger-Nelson problem (which addresses n=2n=2). Frankl and Wilson [FrWi81] proved exponential growth:

χ(Gn)≥(1+o(1))1.2n.\chi(G_n) \geq (1+o(1))1.2^n.

This was improved to

χ(Gn)≥(1.239⋯+o(1))n\chi(G_n)\geq (1.239\cdots+o(1))^n

by Raigorodsky [Ra00]. The trivial colouring (by tiling with cubes) gives

χ(Gn)≤(2+n)n.\chi(G_n) \leq (2+\sqrt{n})^n.

Larman and Rogers [LaRo72] improved this to

χ(Gn)≤(3+o(1))n,\chi(G_n) \leq (3+o(1))^n,

and conjecture the truth may be (23/2+o(1))n(2^{3/2}+o(1))^n. (Note that 23/2≈2.8282^{3/2}\approx 2.828.) Prosanov [Pr20] has given an alternative proof of this upper bound.

See also [508], [705], and [706].

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The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #704 link

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