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We define $ex_{_\\mathcal{P}}(n,\\mathcal{F})$ to be the maximum number of edges in an $\\mathcal{F}$-free planar graph on $n $ vertices. Dowden obtained the tight bounds $ex_{_\\mathcal{P}}(n,C_4)\\leq15(n-2)/7$ for all $n\\geq4$ and $ex_{_\\mathcal{P}}(n,C_5)\\leq(12n-33)/5$ for all $n\\geq1"},"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":"1711.01614","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-05T16:50:52Z","cross_cats_sorted":[],"title_canon_sha256":"28437ecd194d3a39eda4acbab0984bd587ab7f0bc7217a68823bf87bd4a70a32","abstract_canon_sha256":"484928a86f40a8e1ae2733cbf62d0fc55c16c9e59d7b6448b5fb4f575adcb8f2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:55.961191Z","signature_b64":"nDHIOBe6EcdUe6aZOP1g7QDKxOM3UPNu7Ds521G0IJIcCpir7o4Cgl98m2A/aO7eXcBXiZI+UPG5tdYwQXz2Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1078e69236beabb43751d5b26c32d2e8dac96e3089ab4775df58e4cf7e0967b3","last_reissued_at":"2026-05-17T23:52:55.960551Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:55.960551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extremal Theta-free planar graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yongtang Shi, Yongxin Lan, Zi-Xia Song","submitted_at":"2017-11-05T16:50:52Z","abstract_excerpt":"Given a family $\\mathcal{F}$, a graph is $\\mathcal{F}$-free if it does not contain any graph in $\\mathcal{F}$ as a subgraph. 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Dowden obtained the tight bounds $ex_{_\\mathcal{P}}(n,C_4)\\leq15(n-2)/7$ for all $n\\geq4$ and $ex_{_\\mathcal{P}}(n,C_5)\\leq(12n-33)/5$ for all $n\\geq1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.01614","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":"1711.01614","created_at":"2026-05-17T23:52:55.960664+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.01614v3","created_at":"2026-05-17T23:52:55.960664+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.01614","created_at":"2026-05-17T23:52:55.960664+00:00"},{"alias_kind":"pith_short_12","alias_value":"CB4ONERWX2V3","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"CB4ONERWX2V3IN2R","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"CB4ONERW","created_at":"2026-05-18T12:31:10.602751+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/CB4ONERWX2V3IN2R2WZGYMWS5D","json":"https://pith.science/pith/CB4ONERWX2V3IN2R2WZGYMWS5D.json","graph_json":"https://pith.science/api/pith-number/CB4ONERWX2V3IN2R2WZGYMWS5D/graph.json","events_json":"https://pith.science/api/pith-number/CB4ONERWX2V3IN2R2WZGYMWS5D/events.json","paper":"https://pith.science/paper/CB4ONERW"},"agent_actions":{"view_html":"https://pith.science/pith/CB4ONERWX2V3IN2R2WZGYMWS5D","download_json":"https://pith.science/pith/CB4ONERWX2V3IN2R2WZGYMWS5D.json","view_paper":"https://pith.science/paper/CB4ONERW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.01614&json=true","fetch_graph":"https://pith.science/api/pith-number/CB4ONERWX2V3IN2R2WZGYMWS5D/graph.json","fetch_events":"https://pith.science/api/pith-number/CB4ONERWX2V3IN2R2WZGYMWS5D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CB4ONERWX2V3IN2R2WZGYMWS5D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CB4ONERWX2V3IN2R2WZGYMWS5D/action/storage_attestation","attest_author":"https://pith.science/pith/CB4ONERWX2V3IN2R2WZGYMWS5D/action/author_attestation","sign_citation":"https://pith.science/pith/CB4ONERWX2V3IN2R2WZGYMWS5D/action/citation_signature","submit_replication":"https://pith.science/pith/CB4ONERWX2V3IN2R2WZGYMWS5D/action/replication_record"}},"created_at":"2026-05-17T23:52:55.960664+00:00","updated_at":"2026-05-17T23:52:55.960664+00:00"}