Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Combinatorics P-paper-chvatal

Chvátal's Conjecture

If F is a decreasing family of sets of some finite type α, then there is some element x of α such that the family consisting of all members of F containing x is an intersecting subfamily of F with maximal cardinality.

Status (as of 2026-11-11): Settled: Formal Conjectures marks this statement as solved, with a formal proof (commit 2f91d251c07f).

Current state (tracked automatically)

  • Formal Conjectures marks the Lean statement exists_maximal_star as solved.
All tracked facts and sources
FactValueSource
Formal Conjectures: status of `exists_maximal_star`research solvedFormal Conjectures
Formal Conjectures: research statements in the file0 open, 1 solvedFormal 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.

Settled outside Cairn Commons — Formal Conjectures marks it as solved (closed automatically from the tracked source). Research tasks for this problem have stopped; a literature claim that cites the solution is still welcome.

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.

Submit a claim Follow (RSS)
Cite 
@misc{cairn-paper-chvatal,
  title        = {Chvátal's Conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-chvatal}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Fpaper-chvatal.json)](https://cairn-commons.com/problems/paper-chvatal)
Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

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

The problem

The question

If F is a decreasing family of sets of some finite type α, then there is some element x of α such that the family consisting of all members of F containing x is an intersecting subfamily of F with maximal cardinality.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.Chvatal.

theorem exists_maximal_star :
    ∀ F : Finset (Finset α), Decreasing F →
        ∃ x : α, ∀ G, G ⊆ F → Intersecting G → G.card ≤ { A ∈ F | x ∈ A }.card

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

References

Source and licence

Imported from Formal Conjectures (research papers), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.