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We show that for every $\\ell\\ge 5$, $\\ell$ not divisible by $3$, the extremal number is\n  $\n  ex\\left(C_\\ell^-,n\\right)=\\tfrac1{24}n^3+O(n\\ln n)=\\left(\\tfrac14+o(1)\\right){n\\choose 3}.\n  $\n  We determine the extremal graph up to $O(n)$ edge edits."},"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":"2409.14257","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-09-21T22:46:43Z","cross_cats_sorted":[],"title_canon_sha256":"4726a4440611a07c03a75428143f09f0320c8947ac8e509af65dc4571f4d6830","abstract_canon_sha256":"360799bfc09525b7eae299a01294a76978e29bc7727778db39b2c286404cb37e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:40:42.551160Z","signature_b64":"viD0lmcGhIxr1AE3spuKopaY7UCK8I9F0Julj2nqiCN5MgdixbYUhdB0GO7IH5vxM3p8GjPOy+eGUOht4rQhCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b93b2856d5714906829a75c94548f329096da676298e3302d1c3744e3baffffe","last_reissued_at":"2026-07-05T09:40:42.550652Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:40:42.550652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Hypergraph Tur\\'{a}n Densities of Tight Cycles Minus an Edge","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bernard Lidicky, Connor Mattes, Florian Pfender","submitted_at":"2024-09-21T22:46:43Z","abstract_excerpt":"A tight $\\ell$-cycle minus an edge $C_\\ell^-$ is the $3$-graph on the vertex set $[\\ell]$, where any three consecutive vertices in the string $123\\ldots\\ell 1$ form an edge. We show that for every $\\ell\\ge 5$, $\\ell$ not divisible by $3$, the extremal number is\n  $\n  ex\\left(C_\\ell^-,n\\right)=\\tfrac1{24}n^3+O(n\\ln n)=\\left(\\tfrac14+o(1)\\right){n\\choose 3}.\n  $\n  We determine the extremal graph up to $O(n)$ edge edits."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.14257","kind":"arxiv","version":2},"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/2409.14257/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":"2409.14257","created_at":"2026-07-05T09:40:42.550707+00:00"},{"alias_kind":"arxiv_version","alias_value":"2409.14257v2","created_at":"2026-07-05T09:40:42.550707+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.14257","created_at":"2026-07-05T09:40:42.550707+00:00"},{"alias_kind":"pith_short_12","alias_value":"XE5SQVWVOFEQ","created_at":"2026-07-05T09:40:42.550707+00:00"},{"alias_kind":"pith_short_16","alias_value":"XE5SQVWVOFEQNAU2","created_at":"2026-07-05T09:40:42.550707+00:00"},{"alias_kind":"pith_short_8","alias_value":"XE5SQVWV","created_at":"2026-07-05T09:40:42.550707+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.22828","citing_title":"Tur\\'an numbers of $4$-uniform tight even cycles minus one edge","ref_index":20,"is_internal_anchor":false},{"citing_arxiv_id":"2411.01782","citing_title":"The Tur\\'an Density of 4-Uniform Tight Cycles","ref_index":20,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XE5SQVWVOFEQNAU2OXEUKSHTFE","json":"https://pith.science/pith/XE5SQVWVOFEQNAU2OXEUKSHTFE.json","graph_json":"https://pith.science/api/pith-number/XE5SQVWVOFEQNAU2OXEUKSHTFE/graph.json","events_json":"https://pith.science/api/pith-number/XE5SQVWVOFEQNAU2OXEUKSHTFE/events.json","paper":"https://pith.science/paper/XE5SQVWV"},"agent_actions":{"view_html":"https://pith.science/pith/XE5SQVWVOFEQNAU2OXEUKSHTFE","download_json":"https://pith.science/pith/XE5SQVWVOFEQNAU2OXEUKSHTFE.json","view_paper":"https://pith.science/paper/XE5SQVWV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2409.14257&json=true","fetch_graph":"https://pith.science/api/pith-number/XE5SQVWVOFEQNAU2OXEUKSHTFE/graph.json","fetch_events":"https://pith.science/api/pith-number/XE5SQVWVOFEQNAU2OXEUKSHTFE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XE5SQVWVOFEQNAU2OXEUKSHTFE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XE5SQVWVOFEQNAU2OXEUKSHTFE/action/storage_attestation","attest_author":"https://pith.science/pith/XE5SQVWVOFEQNAU2OXEUKSHTFE/action/author_attestation","sign_citation":"https://pith.science/pith/XE5SQVWVOFEQNAU2OXEUKSHTFE/action/citation_signature","submit_replication":"https://pith.science/pith/XE5SQVWVOFEQNAU2OXEUKSHTFE/action/replication_record"}},"created_at":"2026-07-05T09:40:42.550707+00:00","updated_at":"2026-07-05T09:40:42.550707+00:00"}