{"paper":{"title":"Minimum density of union-closed families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Igor Balla","submitted_at":"2011-06-02T06:15:54Z","abstract_excerpt":"Let F be a finite union-closed family of sets whose largest set contains n elements. In \\cite{Wojcik92}, Wojcik defined the density of F to be the ratio of the average set size of F to n and conjectured that the minimum density over all union-closed families whose largest set contains n elements is (1 + o(1))\\log_2(n)/(2n) as n approaches infinity. We use a result of Reimer \\cite{Reimer03} to show that the density of F is always at least log_2(n)/(2n), verifying Wojcik's conjecture. As a corollary we show that for n \\geq 16, some element must appear in at least \\sqrt{(\\log_2(n))/n}(|F|/2) sets"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.0369","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"}