Union-Closed Sets Conjecture

Definitions and mathematical statement

Let $\mathcal F$ be a finite family of sets, let $m=|\mathcal F|$, and let $U=\bigcup_{S\in\mathcal F}S$. Union-closure means \[ \forall A,B\in\mathcal F,\qquad A\cup B\in\mathcal F. \] For $x\in U$, define its frequency count by \[ d_{\mathcal F}(x)=|\{S\in\mathcal F:x\in S\}|. \] The literal stored question is whether every such union-closed family satisfies \[ \exists x\in U,\qquad 2d_{\mathcal F}(x)\ge|\mathcal F|. \] The supplied literature triage records that this exact formulation is false for $\mathcal F=\{\varnothing\}$: it is finite and union-closed, while $U=\varnothing$ provides no candidate element. If an empty family is also permitted by "finite family," it likewise has empty union; the record does not separately state nonemptiness.

The intended nondegenerate Frankl union-closed sets conjecture is the same implication subject to $\mathcal F\ne\varnothing$ and $\mathcal F\ne\{\varnothing\}$, equivalently $U\ne\varnothing$. In the customary finite-ground-set version one additionally takes all members to be finite, so $U$ is finite. The stored sentence itself specifies only finiteness of the family, not of its members; that scope should be checked when aligning the record with the named conjecture and cited theorems. The nondegenerate conjecture is a separately identified intended target, not a silent replacement of the literal false statement.

According to the literature triage checked 17 August 2026, Gilmer proved the first absolute positive frequency constant for the nondegenerate formulation, and Yu's peer-reviewed information-theoretic bound guarantees an element with \[ d_{\mathcal F}(x)\ge0.38234\,|\mathcal F|. \] The later Liu preprint proposes an improvement to approximately $0.38271$ only under numerically verified optimization hypotheses. The record does not provide those hypotheses, and the improvement is not asserted as an unconditional theorem. Neither bound equals the conjectured $1/2$.

The registry's main label is open and its background describes the intended conjecture as unresolved. The nested triage classifies the literal formulation as solved in the literature because of the stated degeneracy, while retaining the corrected nondegenerate conjecture as open. This source-specific distinction must be preserved.

Short English statement

Frankl's intended conjecture says that a nonempty finite union-closed family with nonempty union has an element in at least half its sets. The stored sentence omits the nondegeneracy condition and is refuted by the family containing only the empty set; the supplied literature still treats the intended conjecture as open.

Sources

• Local UnsolvedMath registry, permanent ID $1491$, source code catalogue entry, "Union-Closed Sets Conjecture," including the supplied literature triage checked 17 August 2026.
• J. Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055 (2022), https://arxiv.org/abs/2211.09055; cited for the nondegeneracy hypothesis and first absolute frequency bound.
• L. Yu, Dimension-Free Bounds for the Union-Closed Sets Conjecture, Entropy 25 (2023), 767, https://doi.org/10.3390/e25050767; cited for the rigorous $0.38234$ bound.
• J. Liu, Improving the Lower Bound for the Union-closed Sets Conjecture via Conditionally IID Coupling, arXiv:2306.08824 (2023), https://arxiv.org/abs/2306.08824; cited for the approximately $0.38271$ improvement conditional on numerically verified optimization hypotheses.