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Let $F $ be a fixed graph with $ \\chi(F) \\geq 3 $. A forest $H$ is called a linear forest if all components of $H$ are paths. In this paper, we determined the exact value of $ex(n, \\{H, F\\}) $ for a fixed graph $F$ with $\\chi(F)\\geq 3$ and a linear forest $H$ with at least $2$ components and each component with size at least $3$."},"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":"2507.11034","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-15T06:56:13Z","cross_cats_sorted":[],"title_canon_sha256":"60617ff51445f595d3bc92edd22e46758139cd2cb399500b5541855dbcaf95f7","abstract_canon_sha256":"f5f71dc0895de0d058199cdcdbf9921fc874476728994f57b8e194602a13390d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:37:30.744436Z","signature_b64":"2RiHBUGAXSh4XLjkVuc/8uhdMxvc2KSiFQfKVNbXTEMRUVITIq1DARKYpp0IK6c+CALeujHv9Vov9CI2iHAHDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3da9980a46318f0b232234ef3f72813c7a13a314602b1c397b2d5c9f5ba3a1e8","last_reissued_at":"2026-07-05T11:37:30.743862Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:37:30.743862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tur\\'an type problems for a fixed graph and a linear forest","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Haixiang Zhang, Mei Lu, Xiamiao Zhao","submitted_at":"2025-07-15T06:56:13Z","abstract_excerpt":"Let $\\mathscr{F}$ be a family of graphs. 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