{"paper":{"title":"A lemma on a finite union-closed family of finite sets and its applications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Qian-Qian Zhou, Yi-Ding Shi, Ze-Chun Hu","submitted_at":"2025-07-15T05:56:02Z","abstract_excerpt":"Suppose that $\\mathscr{F}$ is a finite union-closed family of sets with $\\cup_{A\\in \\mathscr{F}}A=\\{1,2,\\ldots,m\\}$ and $m\\geq 2$. Fix $i\\in \\{1,2,\\ldots,m\\}$ and denote $\\mathscr{G}:=\\{A\\backslash \\{i\\}: A\\in \\mathscr{F}\\}$. For $j\\in \\{1,2,\\ldots,m\\}\\backslash\\{i\\}$, let $\\mathscr{G}_j:=\\{A\\in\\mathscr{G}: j\\in A\\}$ and $\\mathscr{F}_j:=\\{A\\in\\mathscr{F}: j\\in A\\}$. In this note, we will prove a lemma which says that if $\\frac{|\\mathscr{G}_j|}{|\\mathscr{G}|}\\geq c\\,(c\\in (0,1])$, then $\\frac{|\\mathscr{F}_j|}{|\\mathscr{F}|}\\geq \\frac{1}{1+2(1-c)/c}$. Several applications of this lemma will be g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.11008","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.11008/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}