{"data":{"id":"union-closed-frankl-1979","statement_oneline":"In any finite union-closed family of sets (other than the empty family), some element belongs to at least half the sets.","statement_full_latex":"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 $\\mathcal{F}$.","source":{"title":"Union-closed sets conjecture (Frankl)","arxiv":null,"url":"https://en.wikipedia.org/wiki/Union-closed_sets_conjecture","year":1979,"area":"math.CO"},"renown":{"score":5,"rationale":"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":{"score":1,"rationale":"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.","subscores":{"finite_witness":4,"oracle_cost":3,"freshness":1,"seedability":1}},"status":"open","last_verified_open":"2026-01-15","added":"2026-07-24","effective_status":"open","claims":[],"status_events":[{"id":"seed-union-closed-frankl-1979","conjecture_id":"union-closed-frankl-1979","status":"open","date":"2026-07-24T00:00:00.000Z","note":"Loaded from seed data."}],"resolutions":[]},"notice":"Status is community- and machine-tracked and may lag. Verify independently before investing effort."}