The Erdős–Ulam problem

The Erdős–Ulam problem (Ulam 1945). Is there a dense subset of the plane whose points are at pairwise rational distances? Conjecturally, no such set exists.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement erdos_ulam_problem as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `erdos_ulam_problem`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 2 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): 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-ulam-problem,
  title        = {The Erdős–Ulam problem},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-ulam-problem}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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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

The Erdős–Ulam problem (Ulam 1945).

Is there a dense subset of the plane whose points are at pairwise rational distances? Conjecturally, no such set exists. Solymosi and de Zeeuw [SdZ10] proved that if a rational-distance set has infinitely many points on a line or circle, all but finitely many points lie on that line or circle.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.ErdosUlamProblem.

theorem erdos_ulam_problem :
    ∃ S : Set (EuclideanSpace ℝ (Fin 2)), Dense S ∧ PairwiseRationalDistances S

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

References

Wikipedia
[Ul60] Ulam, S. M. (1960). A Collection of Mathematical Problems. Interscience, p. 40.
[SdZ10] Solymosi, J. and de Zeeuw, F. (2010). "On a question of Erdős and Ulam." Discrete Comput. Geom. 43, pp. 393--401. arXiv:0806.3095
[Ta14] Tao, T. (2014). "The Erdős–Ulam problem, varieties of general type, and the Bombieri–Lang conjecture." Blog post, terrytao.wordpress.com
[AE45] Anning, N. H. and Erdős, P. (1945). "Integral distances." Bull. Amer. Math. Soc. 51, pp. 598--600.

Source and licence

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