{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1994:VLGEQ6XZWO3LP4GHF3XCKJXMWW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"857747e1a4efd62d9a2bd603f0f9a2858354aa28b7195d222a6d026914b5ae7c","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"1994-09-16T00:00:00Z","title_canon_sha256":"29563210986573c49f2203b65977123614b79ded34c6c90533a374eba518aa48"},"schema_version":"1.0","source":{"id":"math/9409215","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9409215","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/9409215v1","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9409215","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"pith_short_12","alias_value":"VLGEQ6XZWO3L","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"VLGEQ6XZWO3LP4GH","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"VLGEQ6XZ","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:e5edd2a870f54e6db3dcb38cf84ff8c814aee210f5bd5c532b7521e70d0516b0","target":"graph","created_at":"2026-05-18T01:05:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let G be a graph with vertices V and edges E. Let F be the union-closed family of sets generated by E. Then F is the family of subsets of V without isolated points. Theorem: There is an edge e belongs to E such that |{U belongs to F | e belongs to U}| =< 1/2|F|. This is equivalent to the following assertion: If H is a union-closed family generated by a family of sets of maximum degree two, then there is an $x$ such that |{U belongs to H | x belongs to U}| > 1/2|H|. This is a special case of the union-closed sets conjecture. To put this result in perspective, a brief overview of research on the","authors_text":"Emanuel Knill","cross_cats":[],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"1994-09-16T00:00:00Z","title":"Graph generated union-closed families of sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9409215","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3e2631acddbcdaeac1ef93593c371274bf84dba98e329b7f09dde0ac268a824e","target":"record","created_at":"2026-05-18T01:05:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"857747e1a4efd62d9a2bd603f0f9a2858354aa28b7195d222a6d026914b5ae7c","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"1994-09-16T00:00:00Z","title_canon_sha256":"29563210986573c49f2203b65977123614b79ded34c6c90533a374eba518aa48"},"schema_version":"1.0","source":{"id":"math/9409215","kind":"arxiv","version":1}},"canonical_sha256":"aacc487af9b3b6b7f0c72eee2526ecb5b1fb89df60327875a867f680d2df63bb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aacc487af9b3b6b7f0c72eee2526ecb5b1fb89df60327875a867f680d2df63bb","first_computed_at":"2026-05-18T01:05:51.096878Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:51.096878Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mHOtOQN2BF2WuXuojg9I7hrSF/Ko8AKkKudbf0c6ppgs7OJCsS5U35pxnsTQPMtq/mNdA44ZPobeiaMs7KIuDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:51.097350Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9409215","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e2631acddbcdaeac1ef93593c371274bf84dba98e329b7f09dde0ac268a824e","sha256:e5edd2a870f54e6db3dcb38cf84ff8c814aee210f5bd5c532b7521e70d0516b0"],"state_sha256":"a602e6661dd07b15e455fe50e6259387e063f9b54129a1f9583954b38140c67f"}