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In order to do so, he showed that all union-closed families satisfy a particular condition, which in turn implies the preceding bound. Here, answering a question raised in the context of T. Gowers' polymath project on the union-closed sets conjecture, we show t"},"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":"1704.07022","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-24T03:08:08Z","cross_cats_sorted":[],"title_canon_sha256":"12287021aa6518d6f50a1ba854c37d7f2c462218eb2e4b8ad60c4ac47e9031b4","abstract_canon_sha256":"ed5cedd5286ef7863caabc183fa68ce2c50b575758d12befd8cb15160f41f3e1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:54.967865Z","signature_b64":"62QCkB+tYjrSwzIK3Co9oAfAwcoFht5hBW3tgSAAu84IuecULMG0AD/SEeiqTIH4y3c8A81Oy8VLZMTSruhAAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c598e9df960152ec092543d8c494d058a3ba4632ed81de624a4eb4a442352fe","last_reissued_at":"2026-05-18T00:45:54.967218Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:54.967218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Note on the union-closed sets conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abigail Raz","submitted_at":"2017-04-24T03:08:08Z","abstract_excerpt":"The union-closed sets conjecture states that if a family of sets $\\mathcal{A} \\neq \\{\\emptyset\\}$ is union-closed, then there is an element which belongs to at least half the sets in $\\mathcal{A}$. 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