Erdős (Erdos) Problem #704: status in 2026

Erdős Problem #704

Let G_n be the unit distance graph in ℝ^n, with two vertices joined by an edge if and only if the distance between them is 1. Does lim_n→∞χ(G_n)^1/n exist? This generalises the Hadwiger–Nelson problem (Erdős Problem 508), the case n = 2.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #704 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_704.parts.ii as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #704 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_704.parts.ii`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 7 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-704,
  title        = {Erdős Problem #704},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-704}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Let $G_{n}$ be the unit distance graph in $R^{n}$ , with two vertices joined by an edge if and only if the distance between them is $1$ . Does $n \to \infty lim χ (G_{n})^{1/ n}$ exist?

This generalises the Hadwiger–Nelson problem (Erdős Problem 508), the case $n = 2$ . The chromatic number is finite (see Erdos704.erdos_704.variants.cube_colouring).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«704».

theorem erdos_704.parts.ii :
    ∃ L : ℝ, Tendsto
      (fun n : ℕ ↦ ((unitDistanceGraph n).chromaticNumber.toNat : ℝ) ^ (1 / (n : ℝ)))
      atTop (𝓝 L)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/704. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/704
[Da26] Davies, J., Measurable obstructions for unmeasurable colourings. Preprint (2026). arXiv:2610.12301
[Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
[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, R., *A new proof of the Larman-Rogers upper bound for the chromatic number of the Euclidean space*. Discrete Appl. Math. (2020), 115-120.
[Ra00] Raigorodskii, A. M., On the chromatic number of a space. Uspekhi Mat. Nauk (2000), 147-148.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.