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Level A · Machine-checkable Hard Combinatorics P-erdos-539

Erdős Problem #539

Let h(n) be maximal such that, for any set A⊆ ℕ of size n, the set a/(a,b): a,b∈ Ahas size at least h(n). Estimate h(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. 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-539,
  title        = {Erdős Problem #539},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-539}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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

Let be maximal such that, for any set of size , the sethas size at least . Estimate .

In this problem, a function is defined maximally by a specified counting property.

The problem asks to estimate . This has been interpreted here as asking for . The principal version includes answer(sorry) for an unknown function. On the other hand, the best known upper bound is and the best known lower bound is so we also provide these candidates as variants. Moreover, it suffices to show and respectively for each, so further variants are provided for those.

In the source paper [Er73], Erdős also remarks that it should not be too difficult to determine . This does not appear on the website, and it is not clear whether this remains open, but we include it here either way.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«539». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_539 :
    (fun n ↦ (cofactorThreshold n : ℝ)) =Θ[atTop] (answer(sorry) : 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/539. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/539
  • [GR99] Granville, A., & Roesler, F. (1999). _The Set of Differences of a Given Set_. The American Mathematical Monthly, 106(4), 338–344.
  • [Er73] Erdős, P., _Problems and results on combinatorial number theory_. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.
  • [Sc+26] Schmitt, J., Gehrunger, T., Dekoninck, J., Bérczi, G., Kreitner, U., Price, L., & Holmes, D. (2026). _ProofCouncil: An LLM Agent for Solving Open Mathematical Problems_. arXiv:2607.09474, Appendix A, Theorem A.1.

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.