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In this paper, we show that for any $m$ and $t\\ge 2m-3\\ge3$, \\[\\text{ex}(n,K_{m},K_{2,t})=\\Theta(n^{\\frac{3}{2}}).\\] This result improves some results of Alon and Shikhelman (J. Combin. Theory Ser. B, 121:146-172, 2016). We also study the $k$-partite $K_{s,t}$-free graph, we show that for any $k\\ge3$ and $t\\ge(k-1)(s"},"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":"1903.03233","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-08T00:54:00Z","cross_cats_sorted":[],"title_canon_sha256":"cae1e953930ba07e6fce6d17ecdeef94713d585fd218b08bc7554acd151e6c17","abstract_canon_sha256":"ab8984d9fe026e9417ab4161b30e88d69bcdaf9196c7a002750a6bb4c3b48ec2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:49.865243Z","signature_b64":"0NEqaSJo7QEOHImsputNS/nqNDDakei0J0ZNMkoOjtN/aTcj8KZpNWvklkar1zDeEop8klYeD5OSbZ/OOO+CCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d80cf7b910a407b518bd2697a25008b984015f37c36e2a15905022ba53574a67","last_reissued_at":"2026-05-17T23:49:49.864763Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:49.864763Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some extremal results on K_{s,t}-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gennian Ge, Tao Zhang","submitted_at":"2019-03-08T00:54:00Z","abstract_excerpt":"For graphs $H$ and $F$, let $\\text{ex}(n,H,F)$ be the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices. 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