Erdős (Erdos) Problem #174: status in 2026

Erdős Problem #174

A finite set A⊂ ℝ^n is called Ramsey if, for any k≥ 1, there exists some d=d(A,k) such that in any k-colouring of ℝ^d there exists a monochromatic copy of A. Characterise the Ramsey sets in ℝ^n. A copy of A is a set congruent to A, as in [EGMRSS73, p. 344].

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 #174 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_174 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #174 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_174`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	3 open, 4 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-174,
  title        = {Erdős Problem #174},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-174}},
  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

Nobody has worked on this problem here yet

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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

A finite set $A \subset R^{n}$ is called Ramsey if, for any $k \geq 1$ , there exists some $d = d (A, k)$ such that in any $k$ -colouring of $R^{d}$ there exists a monochromatic copy of $A$ . Characterise the Ramsey sets in $R^{n}$ .

A copy of $A$ is a set congruent to $A$ , as in [EGMRSS73, p. 344].

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«174».

theorem erdos_174 (n : ℕ) :
    {A : Set (EuclideanSpace ℝ (Fin n)) | IsRamsey A} = erdos_174.answer n

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/174. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_174.variants.graham — Graham has conjectured that every spherical set is Ramsey.
erdos_174.variants.leader_russell_walters — Leader, Russell, and Walters [LRW12] have alternatively conjectured that a set is Ramsey if and only if it is 'subtransitive': it can be embedded in some…

References

erdosproblems.com/174
[EGMRSS73] Erdős, P. and Graham, R. L. and Montgomery, P. and Rothschild, B. L. and Spencer, J. and Straus, E. G., Euclidean Ramsey Theorems I. J. Comb. Th. A (1973), 341-363. PDF
[Er75f] Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. PDF
[FrRo90] Frankl, P. and Rödl, V., A partition property of simplices in Euclidean space. J. Amer. Math. Soc. (1990), 1-7.
[Kr92] Kříž, Igor, All trapezoids are Ramsey. Discrete Math. (1992), 59-62.
[LRW12] Leader, Imre and Russell, Paul A. and Walters, Mark, *Transitive sets in Euclidean Ramsey theory*. J. Combin. Theory Ser. A (2012), 382-396. arXiv:1012.1350

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.