{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:Q2LMB5G73YR324EZYRAGZS4P2C","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":"20cf6ed3df5e825bad3cf2f92cad7eef59cbc7f0cb7b78cb7952ff4a2a63757f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2022-02-15T02:49:26Z","title_canon_sha256":"60f911b41b136f3f590fec7315589820c03532456cb1bcdc0fefad95d7b8f1d9"},"schema_version":"1.0","source":{"id":"2202.07149","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2202.07149","created_at":"2026-07-05T03:56:57Z"},{"alias_kind":"arxiv_version","alias_value":"2202.07149v1","created_at":"2026-07-05T03:56:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2202.07149","created_at":"2026-07-05T03:56:57Z"},{"alias_kind":"pith_short_12","alias_value":"Q2LMB5G73YR3","created_at":"2026-07-05T03:56:57Z"},{"alias_kind":"pith_short_16","alias_value":"Q2LMB5G73YR324EZ","created_at":"2026-07-05T03:56:57Z"},{"alias_kind":"pith_short_8","alias_value":"Q2LMB5G7","created_at":"2026-07-05T03:56:57Z"}],"graph_snapshots":[{"event_id":"sha256:89f02fafde4d994253c03062a5da460c282c8892f929061ef548582f0e612b7f","target":"graph","created_at":"2026-07-05T03:56:57Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2202.07149/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $F$ and $H$ be $k$-uniform hypergraphs. We say $H$ is $F$-saturated if $H$ does not contain a subgraph isomorphic to $F$, but $H+e$ does for any hyperedge $e\\not\\in E(H)$. The saturation number of $F$, denoted $\\mathrm{sat}_k(n,F)$, is the minimum number of edges in a $F$-saturated $k$-uniform hypergraph $H$ on $n$ vertices. Let $C_3^{(3)}$ denote the $3$-uniform loose cycle on $3$ edges. In this work, we prove that\n  \\[\n  \\left(\\frac{4}3+o(1)\\right)n\\leq \\mathrm{sat}_3(n,C_3^{(3)})\\leq \\frac{3}2n+O(1).\n  \\]\n  This is the first non-trivial result on the saturation number for a fixed short ","authors_text":"Alexandr Kostochka, Dara Zirlin, Sean English","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2022-02-15T02:49:26Z","title":"Saturation for the $3$-uniform loose $3$-cycle"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.07149","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":"sha256:8c50ef517b033937e9202d34ed9906bcd0eeec23837e6fb14cf2fe99785de958","target":"record","created_at":"2026-07-05T03:56:57Z","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":"20cf6ed3df5e825bad3cf2f92cad7eef59cbc7f0cb7b78cb7952ff4a2a63757f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2022-02-15T02:49:26Z","title_canon_sha256":"60f911b41b136f3f590fec7315589820c03532456cb1bcdc0fefad95d7b8f1d9"},"schema_version":"1.0","source":{"id":"2202.07149","kind":"arxiv","version":1}},"canonical_sha256":"8696c0f4dfde23bd7099c4406ccb8fd0a278bffe40e902fc727c3166697dcf08","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8696c0f4dfde23bd7099c4406ccb8fd0a278bffe40e902fc727c3166697dcf08","first_computed_at":"2026-07-05T03:56:57.009581Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:56:57.009581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nBGA4CwlPoZNUt5KdCtME1QfPF6YcnNlHDww0Ok4Utk+ysUMehiE5aYq9DeYtUb7DjQbdlPVRfWeCmSR5fcyBg==","signature_status":"signed_v1","signed_at":"2026-07-05T03:56:57.010021Z","signed_message":"canonical_sha256_bytes"},"source_id":"2202.07149","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8c50ef517b033937e9202d34ed9906bcd0eeec23837e6fb14cf2fe99785de958","sha256:89f02fafde4d994253c03062a5da460c282c8892f929061ef548582f0e612b7f"],"state_sha256":"1e67d32463bed90ab3d69af1afbfa9bc8b62a8de3f3a68c927bd469fcf35e57f"}