Fanelli–Moscardelli conjecture on approximate roommates stability

The universal rank-improvement threshold is K(N)=⌈ N/2⌉ for every sufficiently large number of agents.

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-paper-approximately-stable-matching,
  title        = {Fanelli–Moscardelli conjecture on approximate roommates stability},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-approximately-stable-matching}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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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

The universal rank-improvement threshold is $K (N) = ⌈ N /2 ⌉$ for every sufficiently large number of agents.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.ApproximatelyStableMatching.

theorem stability_threshold_tight :
    ∃ N₀ : ℕ, 3 ≤ N₀ ∧
      ∀ N ≥ N₀, Roommates.stabilityThreshold N = (N + 1) / 2

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References

A. Fanelli and L. Moscardelli, Approximately Stable Matching, Section 5.

For complete strict preferences on $N$ agents, let $K (N)$ be the least positive rank-improvement threshold that guarantees a stable matching for every profile. A pair blocks at threshold $k$ when both agents gain at least $k$ positions. Singles have rank $N$ .

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.