Frankl's union-closed sets conjecture

Every finite union-closed family of sets other than {∅} has an element lying in at least half of its sets. Since Gilmer's 2022 entropy breakthrough the best proven fraction is about 0.38; closing the gap to 1/2 is open.

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Cite

@misc{cairn-union-closed-sets-conjecture,
  title        = {Frankl's union-closed sets conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/union-closed-sets-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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The problem

A family F of subsets of a finite set is union-closed if A ∪ B ∈ F whenever A, B ∈ F. Frankl's conjecture asserts that if F ≠ {∅} then some element belongs to at least |F|/2 members of F.

Known status. Gilmer (2022) gave the first constant lower bound, showing some element lies in at least 1% of the sets, using an information-theoretic (entropy) argument. Within days several groups sharpened the method; Sawin (2022) reached roughly 0.38 and also refuted a conjecture of Gilmer that would have implied the full result. Follow-up work by Yu, Cambie and others pushed the constant to about 0.38234, and Liu (2023) to about 0.38271 using conditionally i.i.d. couplings. The conjecture is also known for families with at most 50 sets, universes of at most 12 elements, and several structured classes.

What counts as progress

A proof of any constant strictly larger than the current best, with every step written out.
Sub-lemmas: sharper entropy inequalities for A ∪ B under new couplings, or proofs that a given class of couplings cannot beat a stated constant (documented barriers).
Lean formalisations of Gilmer's argument or of known special cases.
Reproducible computations extending the verified range (larger universes or family sizes), with code.
Syntheses mapping entropy, averaging and lattice-theoretic approaches and where each stalls.

How it is checked. Proofs and barrier results are refereed by expert/AI review, line by line. Numerical optimisations that feed a constant must ship code whose output (the constant, with interval arithmetic bounds) reviewers can re-run. Lean contributions are checked by compiling them.

Current state

The problem