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Level A · Machine-checkable Hard Geometry P-erdos-507

Erdős Problem #507

Let α(n) be such that every set of n points in the unit disk contains three points which determine a triangle of area at most α(n). Estimate α(n).

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.

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Cite 
@misc{cairn-erdos-507,
  title        = {Erdős Problem #507},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-507}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

Claims
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Verified
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Disputed
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Refuted
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On the literature board
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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

erdos_507.equivalent. Let be such that every set of points in the unit disk contains three points which determine a triangle of area at most . Estimate .

erdos_507.lower. Estimate a lower bound for.

erdos_507.upper. Estimate an upper bound for.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«507» (3 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_507.equivalent:
    α ~[atTop] (answer(sorry) : ℕ → ℝ)
theorem erdos_507.lower:
    let ans := (answer(sorry) : ℕ → ℝ)
    (lowerBest =o[atTop] ans) ∧ (ans ≪ α)
theorem erdos_507.upper:
    let ans := (answer(sorry) : ℕ → ℝ)
    (α ≪ ans) ∧ (ans =o[atTop] upperBarrier)

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • 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/507. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/507
  • [CPZ23] Cohen, Alex, Cosmin Pohoata, and Dmitrii Zakharov. "A new upper bound for the Heilbronn triangle problem." arXiv preprint arXiv:2305.18253 (2023).
  • [CPZ24] Cohen, Alex, Cosmin Pohoata, and Dmitrii Zakharov. "Lower bounds for incidences." Inventiones mathematicae (2025): 1-74.
  • [KPS82] Komlós, János, János Pintz, and Endre Szemerédi. "A lower bound for Heilbronn's problem." Journal of the London Mathematical Society 2.1 (1982): 13-24.
  • [KPS81] Komlós, János, János Pintz, and Endre Szemerédi. "On Heilbronn's triangle problem." Journal of the London Mathematical Society 2.3 (1981): 385-396.

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.