Erdős Problem #91 — open problem in geometry

Erdős Problem #91

Suppose A⊂ ℝ^2 has lvert Arvert=n and minimises the number of distinct distances between points in A. Prove that for large n there are at least two (and probably many) such A which are non-similar.

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-91,
  title        = {Erdős Problem #91},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-91}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Suppose $A \subset R^{2}$ has $∣ A ∣ = n$ and minimises the number of distinct distances between points in $A$ . Prove that for large $n$ there are at least two (and probably many) such $A$ which are non-similar.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«91». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_91 :
    (∀ᶠ n : ℕ in atTop, ¬ UniqueMinimizer n) ↔ answer(sorry)

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/91. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

[Er87b] Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177.
[Ko24c] Z. Kovács, A note on Erdős's mysterious remark. arXiv:2412.05190 (2024).
erdosproblems.com/91

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.