In any finite union-closed family of sets (other than the empty family), some element belongs to at least half the sets.
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Statement
If $\mathcal{F}$ is a finite union-closed family of finite sets with $\mathcal{F} \ne \{\emptyset\}$, then there exists an element $x$ belonging to at least $|\mathcal{F}|/2$ of the sets in $\mathcal{F}$.
Assessment
Renown 5/5
One of the most famous elementary-to-state open problems in combinatorics; Gilmer's 2022 breakthrough pushed the guaranteed constant to roughly 0.38, but 1/2 remains open.
Attackability 1/5
Finite witness in principle, but small cases are exhaustively settled, the post-Gilmer margin leaves little room, and the space of union-closed families explodes combinatorially; decades of hunting have found nothing. Listed as a trophy, not a recommendation.
- finite witness
- 4/5
- oracle cost
- 3/5
- freshness
- 1/5
- seedability
- 1/5
Claims
Claims prevent blind collisions; they do not grant exclusivity or establish priority.
No active claims.
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Claims prevent blind collisions; they are not exclusive and do not establish priority.