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Recently, Alon and Frankl~(arXiv2210.15076) determined the exact value of $\\mbox{ex}(n,K_2,\\{K_{k+1},M_{s+1}\\})$, where $K_{k+1}$ and $M_{s+1}$ are complete graph on $k+1$ vertices and matching of size $s+1$, respectively. Soon after, Gerbner~(arXiv2211.03272) continued the study by extending $K_{k+1}$ to general fixed graph $H$. In this paper, we continue the study of the function $\\mbox{ex}(n, T,\\{H,M_{s+1}\\})$ when $T=K_r$ "},"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":"2301.05625","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-01-13T15:51:24Z","cross_cats_sorted":[],"title_canon_sha256":"26efbd6d9c5bd18c6cee6d48b23867d80de44a0c4a6ee96784ac0cc055cfd295","abstract_canon_sha256":"20038c12e8ffab11d50a072d2931e10923825a1b7906180677455eb12758a764"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:06:18.592098Z","signature_b64":"Bg0jzgxtAzTryPJi70nxSRKZOUtcQtwbBiFSmgRZtQVSRJDVXfCfsbLhdXRkTRo13JdxTZGUSszuz/N1kBOXBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8d903db0d93ee4169e6e8e4ba82dfc18d3a572245c8a5dbaa1e87da9c5646ef4","last_reissued_at":"2026-07-05T11:06:18.591559Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:06:18.591559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generalized Tur\\'an problem with bounded matching number","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xinmin Hou, Yue Ma","submitted_at":"2023-01-13T15:51:24Z","abstract_excerpt":"For a graph $T$ and a set of graphs $\\mathcal{H}$, let $\\mbox{ex}(n,T,\\mathcal{H})$ denote the maximum number of copies of $T$ in an $n$-vertex $\\mathcal{H}$-free graph. Recently, Alon and Frankl~(arXiv2210.15076) determined the exact value of $\\mbox{ex}(n,K_2,\\{K_{k+1},M_{s+1}\\})$, where $K_{k+1}$ and $M_{s+1}$ are complete graph on $k+1$ vertices and matching of size $s+1$, respectively. Soon after, Gerbner~(arXiv2211.03272) continued the study by extending $K_{k+1}$ to general fixed graph $H$. 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