The Irving–Leather conjecture on stable matchings

Thurber's §2 attributes to Irving and Leather the conjecture that their construction attains the maximum number of stable matchings at every power-of-two size.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): 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.

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Cite

@misc{cairn-paper-irving-leather-stable-matchings,
  title        = {The Irving–Leather conjecture on stable matchings},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-irving-leather-stable-matchings}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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Disputed

Refuted

On the literature board

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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.

The problem

The question

irving_leather_pow_two. Thurber's §2 attributes to Irving and Leather the conjecture that their construction attains the maximum number of stable matchings at every power-of-two size.

irving_leather_growth_rate. A proposed weaker variant states that $M (n)^{1/ n}$ and $a_{k}^{1/ 2^{k}}$ converge to the same real number. Thomas's recurrence analysis gives approximately $2.280142$ for the second limit. The first sequence is indexed by $n + 1$ to start at positive sizes.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.IrvingLeatherStableMatchings (2 statements).

theorem irving_leather_pow_two (k : ℕ) :
    maxStableMatchings (2 ^ k) = irvingLeatherCount k

theorem irving_leather_growth_rate :
    ∃ c : ℝ,
      Tendsto (fun n : ℕ ↦ Real.rpow (maxStableMatchings (n + 1) : ℝ)
        (1 / ((n : ℝ) + 1))) atTop (𝓝 c) ∧
      Tendsto (fun k : ℕ ↦ Real.rpow (irvingLeatherCount k : ℝ)
        (1 / (2 : ℝ) ^ k)) atTop (𝓝 c)

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. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Concerning the maximum number of stable matchings in the stable marriage problem by E. G. Thurber, §2 and Appendix A.
A recurrence giving a lower bound for stable matchings by Clay Thomas, Proposition 1.

Let $M (n)$ be the maximum number of stable matchings with $n$ participants per side and complete, strict preferences. The Irving–Leather construction has $a_{k}$ stable matchings at size $2^{k}$ .

Source and licence

Imported from Formal Conjectures (research papers), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.