Erdős Problem #89 — open problem in geometry

Erdős Problem #89

Erdős [Er46] asked whether every set of n distinct points in ℝ^2 determines ≫ n/√(log n) many distinct distances.

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-89,
  title        = {Erdős Problem #89},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-89}},
  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

Erdős [Er46] asked whether every set of $n$ distinct points in $R^{2}$ determines $≫ \frac{n}{l o g n}$ many distinct distances.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«89».

theorem erdos_89 :
    (fun (n : ℕ) => n/(n : ℝ).log.sqrt) =O[atTop]
      (fun n => (minimalDistinctDistances ℝ² n : ℝ))

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/89. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/89
[Er46] Erdős, Paul. On sets of distances of $n$ points. Amer. Math. Monthly 53 (1946), 248--250.
[GuKa15] Guth, Larry and Katz, Nets Hawk. On the Erdős distinct distances problem in the plane. Ann. of Math. (2) 181 (2015), 155--190.
[Mo52] Moser, Leo. On the different distances determined by $n$ points. Amer. Math. Monthly 59 (1952), 85--91.

AI disclosure

Lean 4 code in this file was drafted with assistance from OpenAI Codex. The mathematical content and references are the author's own work.

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.