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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1106.0369","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-06-02T06:15:54Z","cross_cats_sorted":[],"title_canon_sha256":"62f7a013fce839bc79f4bd3575f245fc798c935f17f2b21d6c2d954c705be456","abstract_canon_sha256":"1ae64a813a2eedd058e8397f4d0aa19a94ae75881c4757154f56eae0e7bd60d1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:50.560310Z","signature_b64":"1r7c7lygJEOGcUgp+DOaNkp04qGhE1KjgyFn2sg10M7AmxUd1sSyLUDuyoXjlxBnFVmZwdk4efSZmmFoQOTuBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"358adc35b20b9eb42dfa8fa7c7b2926a0223b35e71e9bd13952eab45cd19fe55","last_reissued_at":"2026-05-18T04:20:50.559830Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:50.559830Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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