{"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. A graph $G$ is $\\mathscr{F}$-free if $G$ does not contain any $F\\in \\mathscr{F}$ as a subgraph. The Tur\\'an number, denoted by $ex(n, \\mathscr{F})$, is the maximum number of edges in an $n$-vertex $\\mathscr{F}$-free graph. 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 $3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.11034","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.11034/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}