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Given a linear $r$-graph $H$ and a positive integer $n$, the linear Tur\\'an number $ex_L(n,H)$ is the maximum number of edges in a linear $r$-graph $G$ that does not contain $H$ as a subgraph. For each $\\ell\\geq 3$, let $C^r_\\ell$ denote the $r$-uniform linear cycle of length $\\ell$, which is an $r$-graph with edges $e_1,\\ldots, e_\\ell$ such that $\\forall i\\in [\\ell-1]$, $|e_i\\cap e_{i+1}|=1$, $|e_\\ell\\cap e_1|=1$ and $e_i\\cap e_j=\\emptyset$ for all other pairs $\\{i,j\\},"},"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":"1404.5015","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-04-20T05:41:23Z","cross_cats_sorted":[],"title_canon_sha256":"ff9409ebdb63d0d81ccf659b2bd6c0010ceb0be31cea9ac3abc2c83d71ed2911","abstract_canon_sha256":"b0cbbb5c99e9b2424c0b892f41dd1102f3e25e038133e1a9f2e27e068aa041b6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:28.410294Z","signature_b64":"8vOsLcLHDeXk4pmsXkrBBTuZmw9PlL7DWJRw+mYVuWVTy//QTbc//Y+vkdAJMYoEkaZO4NqJOXTWr2OjQj1fDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f46e2e439ccb359b80ac0d2475330815926693076d842ed82dadc53ebc745a59","last_reissued_at":"2026-05-18T02:53:28.409519Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:28.409519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Linear Turan numbers of r-uniform linear cycles and related Ramsey numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Clayton Collier-Cartaino, Nathan Graber, Tao Jiang","submitted_at":"2014-04-20T05:41:23Z","abstract_excerpt":"An $r$-uniform hypergraph is called an $r$-graph. A hypergraph is linear if every two edges intersect in at most one vertex. Given a linear $r$-graph $H$ and a positive integer $n$, the linear Tur\\'an number $ex_L(n,H)$ is the maximum number of edges in a linear $r$-graph $G$ that does not contain $H$ as a subgraph. For each $\\ell\\geq 3$, let $C^r_\\ell$ denote the $r$-uniform linear cycle of length $\\ell$, which is an $r$-graph with edges $e_1,\\ldots, e_\\ell$ such that $\\forall i\\in [\\ell-1]$, $|e_i\\cap e_{i+1}|=1$, $|e_\\ell\\cap e_1|=1$ and $e_i\\cap e_j=\\emptyset$ for all other pairs $\\{i,j\\},"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.5015","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":""},"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":"1404.5015","created_at":"2026-05-18T02:53:28.409646+00:00"},{"alias_kind":"arxiv_version","alias_value":"1404.5015v2","created_at":"2026-05-18T02:53:28.409646+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.5015","created_at":"2026-05-18T02:53:28.409646+00:00"},{"alias_kind":"pith_short_12","alias_value":"6RXC4Q44ZM2Z","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_16","alias_value":"6RXC4Q44ZM2ZXAFM","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_8","alias_value":"6RXC4Q44","created_at":"2026-05-18T12:28:16.859392+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/6RXC4Q44ZM2ZXAFMBUSHKMYICW","json":"https://pith.science/pith/6RXC4Q44ZM2ZXAFMBUSHKMYICW.json","graph_json":"https://pith.science/api/pith-number/6RXC4Q44ZM2ZXAFMBUSHKMYICW/graph.json","events_json":"https://pith.science/api/pith-number/6RXC4Q44ZM2ZXAFMBUSHKMYICW/events.json","paper":"https://pith.science/paper/6RXC4Q44"},"agent_actions":{"view_html":"https://pith.science/pith/6RXC4Q44ZM2ZXAFMBUSHKMYICW","download_json":"https://pith.science/pith/6RXC4Q44ZM2ZXAFMBUSHKMYICW.json","view_paper":"https://pith.science/paper/6RXC4Q44","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1404.5015&json=true","fetch_graph":"https://pith.science/api/pith-number/6RXC4Q44ZM2ZXAFMBUSHKMYICW/graph.json","fetch_events":"https://pith.science/api/pith-number/6RXC4Q44ZM2ZXAFMBUSHKMYICW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6RXC4Q44ZM2ZXAFMBUSHKMYICW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6RXC4Q44ZM2ZXAFMBUSHKMYICW/action/storage_attestation","attest_author":"https://pith.science/pith/6RXC4Q44ZM2ZXAFMBUSHKMYICW/action/author_attestation","sign_citation":"https://pith.science/pith/6RXC4Q44ZM2ZXAFMBUSHKMYICW/action/citation_signature","submit_replication":"https://pith.science/pith/6RXC4Q44ZM2ZXAFMBUSHKMYICW/action/replication_record"}},"created_at":"2026-05-18T02:53:28.409646+00:00","updated_at":"2026-05-18T02:53:28.409646+00:00"}