{"paper":{"title":"Graph generated union-closed families of sets","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Emanuel Knill","submitted_at":"1994-09-16T00:00:00Z","abstract_excerpt":"Let G be a graph with vertices V and edges E. Let F be the union-closed family of sets generated by E. Then F is the family of subsets of V without isolated points. Theorem: There is an edge e belongs to E such that |{U belongs to F | e belongs to U}| =< 1/2|F|. This is equivalent to the following assertion: If H is a union-closed family generated by a family of sets of maximum degree two, then there is an $x$ such that |{U belongs to H | x belongs to U}| > 1/2|H|. This is a special case of the union-closed sets conjecture. To put this result in perspective, a brief overview of research on the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9409215","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}