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Let $C_k$ denote the cycle of length $k$ and let $\\mathscr C_k=\\{C_3,C_4,\\ldots,C_k\\}$. Some of our main results are the following.\n  (i) We show that $ex(n, C_{2l}, C_{2k}) = \\Theta(n^l)$ for any $l, k \\ge 2$. Moreover, we determine it asymptotically in the following cases: We show that $ex(n,C_4,C_{2k}) = (1+o(1)) \\frac{("},"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":"1712.07079","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-19T17:52:43Z","cross_cats_sorted":[],"title_canon_sha256":"13a4b13fc9c2e6150e701134cf868f4ee4251947ea438f8473b483db86d84f35","abstract_canon_sha256":"6f6a9d50fc728b61f39059afeac8a2007b5f1eaf50454fc94c584c943d145331"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:14.741895Z","signature_b64":"9D8fBa7VG9ooR9qaZdoHyr+Rg6iTIBgHu44Tega04cZApj2rh2YE29tO6Y7Ik4c4kuSg9vAYbrat0T91H7mTDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8fe107807177a7e1e5648c22dfad7ce50be1136efc44957047ef4ee519123543","last_reissued_at":"2026-05-17T23:58:14.741415Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:14.741415Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generalized Tur\\'an problems for even cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abhishek Methuku, D\\'aniel Gerbner, Ervin Gy\\H{o}ri, M\\'at\\'e Vizer","submitted_at":"2017-12-19T17:52:43Z","abstract_excerpt":"Given a graph $H$ and a set of graphs $\\mathcal F$, let $ex(n,H,\\mathcal F)$ denote the maximum possible number of copies of $H$ in an $\\mathcal F$-free graph on $n$ vertices. We investigate the function $ex(n,H,\\mathcal F)$, when $H$ and members of $\\mathcal F$ are cycles. Let $C_k$ denote the cycle of length $k$ and let $\\mathscr C_k=\\{C_3,C_4,\\ldots,C_k\\}$. Some of our main results are the following.\n  (i) We show that $ex(n, C_{2l}, C_{2k}) = \\Theta(n^l)$ for any $l, k \\ge 2$. Moreover, we determine it asymptotically in the following cases: We show that $ex(n,C_4,C_{2k}) = (1+o(1)) \\frac{("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.07079","kind":"arxiv","version":3},"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":"1712.07079","created_at":"2026-05-17T23:58:14.741496+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.07079v3","created_at":"2026-05-17T23:58:14.741496+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.07079","created_at":"2026-05-17T23:58:14.741496+00:00"},{"alias_kind":"pith_short_12","alias_value":"R7QQPADRO6T6","created_at":"2026-05-18T12:31:39.905425+00:00"},{"alias_kind":"pith_short_16","alias_value":"R7QQPADRO6T6DZLE","created_at":"2026-05-18T12:31:39.905425+00:00"},{"alias_kind":"pith_short_8","alias_value":"R7QQPADR","created_at":"2026-05-18T12:31:39.905425+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/R7QQPADRO6T6DZLERQRN7LL44U","json":"https://pith.science/pith/R7QQPADRO6T6DZLERQRN7LL44U.json","graph_json":"https://pith.science/api/pith-number/R7QQPADRO6T6DZLERQRN7LL44U/graph.json","events_json":"https://pith.science/api/pith-number/R7QQPADRO6T6DZLERQRN7LL44U/events.json","paper":"https://pith.science/paper/R7QQPADR"},"agent_actions":{"view_html":"https://pith.science/pith/R7QQPADRO6T6DZLERQRN7LL44U","download_json":"https://pith.science/pith/R7QQPADRO6T6DZLERQRN7LL44U.json","view_paper":"https://pith.science/paper/R7QQPADR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.07079&json=true","fetch_graph":"https://pith.science/api/pith-number/R7QQPADRO6T6DZLERQRN7LL44U/graph.json","fetch_events":"https://pith.science/api/pith-number/R7QQPADRO6T6DZLERQRN7LL44U/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/R7QQPADRO6T6DZLERQRN7LL44U/action/timestamp_anchor","attest_storage":"https://pith.science/pith/R7QQPADRO6T6DZLERQRN7LL44U/action/storage_attestation","attest_author":"https://pith.science/pith/R7QQPADRO6T6DZLERQRN7LL44U/action/author_attestation","sign_citation":"https://pith.science/pith/R7QQPADRO6T6DZLERQRN7LL44U/action/citation_signature","submit_replication":"https://pith.science/pith/R7QQPADRO6T6DZLERQRN7LL44U/action/replication_record"}},"created_at":"2026-05-17T23:58:14.741496+00:00","updated_at":"2026-05-17T23:58:14.741496+00:00"}