{"paper":{"title":"Two Results on Union-Closed Families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ilan Karpas","submitted_at":"2017-08-04T09:58:32Z","abstract_excerpt":"We show that there is some absolute constant $c>0$, such that for any union-closed family $\\mathcal{F} \\subseteq 2^{[n]}$, if \\mbox{$|\\mathcal{F}| \\geq (\\frac{1}{2}-c)2^n$}, then there is some element $i \\in [n]$ that appears in at least half of the sets of $\\mathcal{F}$. We also show that for any union-closed family $\\mathcal{F} \\subseteq 2^{[n]}$, the number of sets which are not in $\\mathcal{F}$ that cover a set in $\\mathcal{F}$ is at most $2^{n-1}$, and provide examples where the inequality is tight."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01434","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"}