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Level A · Machine-checkable Hard Geometry P-erdos-506

Erdős Problem #506

Erdős Problem #506

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-506,
  title        = {Erdős Problem #506},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-506}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

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

The problem

The question

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«506». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_506 (n : ℕ) (hn : 4 ≤ n) :
    IsLeast { k : ℕ | ∃ P : Finset ℝ²,
        P.card = n ∧ ¬ Collinear ℝ (P : Set ℝ²) ∧ ¬ Cospherical (P : Set ℝ²) ∧
        numCircles (P : Set ℝ²) = k }
      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/506. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_506.variants.small_n — The problem appears to remain open for small n: Elliott's answer is established only for n > 393, so the minimum number of circles determined by n points, not…

References

  • erdosproblems.com/506
  • [El67] Elliott, P. D. T. A., On the number of circles determined by points, Acta Math. Acad. Sci. Hungar. (1967), 181–188.
  • [BaBa94] Bálintová, A. and Bálint, V., *On the number of circles determined by points in the Euclidean plane*, Acta Math. Hungar. (1994), 283–289.
  • [PuSm] Purdy and Smith. No reference found.

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.