{"paper":{"title":"Generalized Tur\\'an problems for a matching and long cycles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mei Lu, Xiamiao Zhao","submitted_at":"2024-12-25T09:51:54Z","abstract_excerpt":"Let $\\mathscr{F}$ be a family of graphs. A graph $G$ is $\\mathscr{F}$-free if $G$ does not contain any $F\\in \\mathcal{F}$ as a subgraph. The general Tur\\'an number, denoted by $ex(n, H,\\mathscr{F})$, is the maximum number of copies of $H$ in an $n$-vertex $\\mathscr{F}$-free graph. Then $ex(n, K_2,\\mathscr{F})$, also denote by $ex(n, \\mathscr{F})$, is the Tur\\'an number. Recently, Alon and Frankl determined the exact value of $ex(n, \\{K_{k},M_{s+1}\\})$, where $K_{k}$ and $M_{s+1}$ are a complete graph on $k $ vertices and a matching of size $s +1$, respectively. Then many results were obtained "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.18853","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/2412.18853/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"}