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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 "},"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":"2202.07149","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2022-02-15T02:49:26Z","cross_cats_sorted":[],"title_canon_sha256":"60f911b41b136f3f590fec7315589820c03532456cb1bcdc0fefad95d7b8f1d9","abstract_canon_sha256":"20cf6ed3df5e825bad3cf2f92cad7eef59cbc7f0cb7b78cb7952ff4a2a63757f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:56:57.010021Z","signature_b64":"nBGA4CwlPoZNUt5KdCtME1QfPF6YcnNlHDww0Ok4Utk+ysUMehiE5aYq9DeYtUb7DjQbdlPVRfWeCmSR5fcyBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8696c0f4dfde23bd7099c4406ccb8fd0a278bffe40e902fc727c3166697dcf08","last_reissued_at":"2026-07-05T03:56:57.009581Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:56:57.009581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Saturation for the $3$-uniform loose $3$-cycle","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr Kostochka, Dara Zirlin, Sean English","submitted_at":"2022-02-15T02:49:26Z","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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.07149","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2202.07149/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2202.07149","created_at":"2026-07-05T03:56:57.009643+00:00"},{"alias_kind":"arxiv_version","alias_value":"2202.07149v1","created_at":"2026-07-05T03:56:57.009643+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2202.07149","created_at":"2026-07-05T03:56:57.009643+00:00"},{"alias_kind":"pith_short_12","alias_value":"Q2LMB5G73YR3","created_at":"2026-07-05T03:56:57.009643+00:00"},{"alias_kind":"pith_short_16","alias_value":"Q2LMB5G73YR324EZ","created_at":"2026-07-05T03:56:57.009643+00:00"},{"alias_kind":"pith_short_8","alias_value":"Q2LMB5G7","created_at":"2026-07-05T03:56:57.009643+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Q2LMB5G73YR324EZYRAGZS4P2C","json":"https://pith.science/pith/Q2LMB5G73YR324EZYRAGZS4P2C.json","graph_json":"https://pith.science/api/pith-number/Q2LMB5G73YR324EZYRAGZS4P2C/graph.json","events_json":"https://pith.science/api/pith-number/Q2LMB5G73YR324EZYRAGZS4P2C/events.json","paper":"https://pith.science/paper/Q2LMB5G7"},"agent_actions":{"view_html":"https://pith.science/pith/Q2LMB5G73YR324EZYRAGZS4P2C","download_json":"https://pith.science/pith/Q2LMB5G73YR324EZYRAGZS4P2C.json","view_paper":"https://pith.science/paper/Q2LMB5G7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2202.07149&json=true","fetch_graph":"https://pith.science/api/pith-number/Q2LMB5G73YR324EZYRAGZS4P2C/graph.json","fetch_events":"https://pith.science/api/pith-number/Q2LMB5G73YR324EZYRAGZS4P2C/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Q2LMB5G73YR324EZYRAGZS4P2C/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Q2LMB5G73YR324EZYRAGZS4P2C/action/storage_attestation","attest_author":"https://pith.science/pith/Q2LMB5G73YR324EZYRAGZS4P2C/action/author_attestation","sign_citation":"https://pith.science/pith/Q2LMB5G73YR324EZYRAGZS4P2C/action/citation_signature","submit_replication":"https://pith.science/pith/Q2LMB5G73YR324EZYRAGZS4P2C/action/replication_record"}},"created_at":"2026-07-05T03:56:57.009643+00:00","updated_at":"2026-07-05T03:56:57.009643+00:00"}