{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:WAN4PF2JLNSE4DYQMG27LNT45L","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":"0515de630de9c67bbd756956030d31c9b4476ae6c984686a5e2e9ff55020a783","cross_cats_sorted":["math.CO","math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2004-09-05T14:59:05Z","title_canon_sha256":"42fb3899e87d9f8273ffdd2e8897b786f5727ccf74bd1fada8491b74f6c5aade"},"schema_version":"1.0","source":{"id":"math/0409068","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0409068","created_at":"2026-05-18T04:37:23Z"},{"alias_kind":"arxiv_version","alias_value":"math/0409068v4","created_at":"2026-05-18T04:37:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0409068","created_at":"2026-05-18T04:37:23Z"},{"alias_kind":"pith_short_12","alias_value":"WAN4PF2JLNSE","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"WAN4PF2JLNSE4DYQ","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"WAN4PF2J","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:359e3770acaca209a30b97ff40df3492ac3b40d5eeadd73b380959e24cba458c","target":"graph","created_at":"2026-05-18T04:37:23Z","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":"We solve four out of the six open problems concerning critical cardinalities of topological diagonalization properties involving tau-covers, show that the remaining two cardinals are equal, and give a consistency result concerning this remaining cardinal. Consequently, 21 open problems concerning potential implications between these properties are settled. We also give structural results based on the combinatorial techniques.","authors_text":"Boaz Tsaban, Heike Mildenberger, Saharon Shelah","cross_cats":["math.CO","math.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2004-09-05T14:59:05Z","title":"The combinatorics of tau-covers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0409068","kind":"arxiv","version":4},"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":"sha256:240ceac91d8db18f32b1f804f5b18d26260b6ac543758f3496e99fa554033015","target":"record","created_at":"2026-05-18T04:37:23Z","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":"0515de630de9c67bbd756956030d31c9b4476ae6c984686a5e2e9ff55020a783","cross_cats_sorted":["math.CO","math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2004-09-05T14:59:05Z","title_canon_sha256":"42fb3899e87d9f8273ffdd2e8897b786f5727ccf74bd1fada8491b74f6c5aade"},"schema_version":"1.0","source":{"id":"math/0409068","kind":"arxiv","version":4}},"canonical_sha256":"b01bc797495b644e0f1061b5f5b67ceacb1117e3da16220ac16f8bf73f1b933a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b01bc797495b644e0f1061b5f5b67ceacb1117e3da16220ac16f8bf73f1b933a","first_computed_at":"2026-05-18T04:37:23.165679Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:37:23.165679Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NnD55tK+4zXjmbSwgKgRAqUaCFk+EM+tAWQvTMYSKfc6pOLbOkuMXOJHmU8XMzgYkwI+/CfljPW7/QOnsSBHBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:37:23.166086Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0409068","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:240ceac91d8db18f32b1f804f5b18d26260b6ac543758f3496e99fa554033015","sha256:359e3770acaca209a30b97ff40df3492ac3b40d5eeadd73b380959e24cba458c"],"state_sha256":"81bc1fb7ed8ce4bb8cee6649c23db5ee336786c77cb4b15bb5aa2b869b2f6e6c"}