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We define $ex_{_\\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield $ex_{_\\mathcal{P}}(n,H)=3n-6$ for all $n\\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erd\\H{o}s-Stone Theorem. We then completely determine $ex_{_\\ma"},"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":"1808.01487","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-04T14:41:45Z","cross_cats_sorted":[],"title_canon_sha256":"3a1ed2587c18491622b77125f566986b5b67119cc1697408f40cfc0a4904929c","abstract_canon_sha256":"a3bb2a3822787ebb239300a247c05f883128a8ecbad217a5fc1353c73b32ec7d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:52.812303Z","signature_b64":"2jgT3F8kkJie/l8++38gkHXr0u0hQX/YF9GP/QmCwD+UY1tKWCSXVJ2o8uV8xYciE79bukmgnf/A+uc7Jod8Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8cce220313c2a0c2a84c12a7ac7c71906412c5ffc6de8369dcccc372c94c661","last_reissued_at":"2026-05-18T00:08:52.811449Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:52.811449Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extremal $H$-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":"2018-08-04T14:41:45Z","abstract_excerpt":"Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. 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