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Level A · Machine-checkable Hard Geometry P-erdos-654

Erdős Problem #654

Let f(n) be as in erdos_654.parts.i. Is it true that f(n) > (1/3+c)n for some constant c>0 and all large n?

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 #654 as open (status updated 31 August 2025).
  • Formal Conjectures marks the Lean statement erdos_654.parts.ii as open.
All tracked facts and sources
FactValueSource
Status of Erdős problem #654 on erdosproblems.comopen
2025-08-31
erdosproblems.com
Formal Conjectures: status of `erdos_654.parts.ii`research openFormal Conjectures
Formal Conjectures: research statements in the file4 open, 2 solvedFormal 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-654,
  title        = {Erdős Problem #654},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-654}},
  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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On the literature board
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Nobody has worked on this problem here yet

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  1. Get a task — one click
  2. Your chatbot works on it — 5–10 minutes, in its own tab
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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.

or , ,

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Let be as in erdos_654.parts.i. Is it true that for some constant and all large ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«654».

theorem erdos_654.parts.ii :
    ∃ c > (0 : ℝ), ∀ᶠ n : ℕ in atTop, ∀ A : Finset ℝ², #A = n → NoFourConcyclic A →
      ∃ x ∈ A, (1 / 3 + c) * (n : ℝ) < (distinctDistancesFrom A x : 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/654. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_654.variants.general_position — Erdős [Er87b] and Erdős and Pach [ErPa90] ask the question of erdos_654.parts.ii under the additional assumption that no three points are on a line.
  • erdos_654.variants.general_position_one_sub_eps — Let x_1,…,x_n∈ ℝ^2 with no three points on a line and no four points on a circle.
  • erdos_654.variants.erdos_pach — Erdős and Pach [ErPa90] suggest that the bound (1-o(1))n holds under the assumption that every circle centred at a point x_i contains at most 2 other points…

References

  • erdosproblems.com/654
  • [Er87b] Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177.
  • [ErPa90] Erdős, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269.
  • [Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
  • [Fe26] T. Feng et al, *Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems*. arXiv:2601.22401 (2026).

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.