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Level A · Machine-checkable Hard Number theory P-erdos-302

Erdős Problem #302

Let f(N) be the size of the largest A⊆ 1,…,N such that there are no solutions to 1/a= 1/b+1/c with distinct a,b,c∈ A? Estimate f(N). The colouring version of this is [303], which was solved by Brown and Rödl [BrRo91].

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-302,
  title        = {Erdős Problem #302},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-302}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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

The problem

The question

Let be the size of the largest such that there are no solutions to with distinct ? Estimate .

The colouring version of this is [303], which was solved by Brown and Rödl [BrRo91].

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«302». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_302.parts.i (f : ℕ → ℕ) (hf : ∀ N, IsMaxNoTripleCard N (f N)) :
    Tendsto (fun N : ℕ => (f N : ℝ) / N) atTop (𝓝 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/302. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/302
  • [BrRo91] Brown, Tom C. and Rödl, Voijtech, Monochromatic solutions to equations with unit fractions. Bull. Austral. Math. Soc. (1991), 387-392.
  • [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
  • va25

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.