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We prove that if $s\\in \\{2,3\\}$, $s< a\\leq b$ and $t$ is sufficiently large, then $\\mathrm{ex}(n,K_{a,b},K_{s,t})=\\Theta(n^s)$. The $s=2$, $a=b=3$ case of this result answers a question of Spiro.\n  Proving another conjecture of Spiro, we show that for every graph $F$ with at least one edge, there exist infinitely many real numbers $r$ such that $\\mathrm{ex}(n,F,H)=\\Theta(n^r)$ holds for some graph $H$."},"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":"2606.09801","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-08T17:53:07Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"7108c969af300ef88e49e848d61347949fd8db247fd198c358425b5925c72a69","abstract_canon_sha256":"4082da0b3408dfce2db433ec3a74257dfe4f123962a3b0fce438facaa9099e19"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T02:09:10.595070Z","signature_b64":"W8tq5CWFSWev4urvEblDrC4wUppNAHZZE2N8o/yAIMm97ta9pjffoD4/if4FC342KkWgSnqMbb/lhKfXxnryBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cc04dfd471e4ce866afa5c0fb411ae4b1676ceb9a2e03c7081d8d2c94e59ebb8","last_reissued_at":"2026-06-09T02:09:10.594362Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T02:09:10.594362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the generalized Tur\\'an number of complete bipartite graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Liana Yepremyan, Oliver Janzer, Sean Longbrake","submitted_at":"2026-06-08T17:53:07Z","abstract_excerpt":"For graphs $F$ and $H$, the generalized Tur\\'an number $\\mathrm{ex}(n,F,H)$ denotes the maximum number of copies of $F$ in an $H$-free graph on $n$ vertices. We prove that if $s\\in \\{2,3\\}$, $s< a\\leq b$ and $t$ is sufficiently large, then $\\mathrm{ex}(n,K_{a,b},K_{s,t})=\\Theta(n^s)$. The $s=2$, $a=b=3$ case of this result answers a question of Spiro.\n  Proving another conjecture of Spiro, we show that for every graph $F$ with at least one edge, there exist infinitely many real numbers $r$ such that $\\mathrm{ex}(n,F,H)=\\Theta(n^r)$ holds for some graph $H$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.09801","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/2606.09801/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.09801","created_at":"2026-06-09T02:09:10.594471+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.09801v1","created_at":"2026-06-09T02:09:10.594471+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.09801","created_at":"2026-06-09T02:09:10.594471+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZQCN7VDR4THI","created_at":"2026-06-09T02:09:10.594471+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZQCN7VDR4THIM2X2","created_at":"2026-06-09T02:09:10.594471+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZQCN7VDR","created_at":"2026-06-09T02:09:10.594471+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZQCN7VDR4THIM2X2LQH3IENOJM","json":"https://pith.science/pith/ZQCN7VDR4THIM2X2LQH3IENOJM.json","graph_json":"https://pith.science/api/pith-number/ZQCN7VDR4THIM2X2LQH3IENOJM/graph.json","events_json":"https://pith.science/api/pith-number/ZQCN7VDR4THIM2X2LQH3IENOJM/events.json","paper":"https://pith.science/paper/ZQCN7VDR"},"agent_actions":{"view_html":"https://pith.science/pith/ZQCN7VDR4THIM2X2LQH3IENOJM","download_json":"https://pith.science/pith/ZQCN7VDR4THIM2X2LQH3IENOJM.json","view_paper":"https://pith.science/paper/ZQCN7VDR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.09801&json=true","fetch_graph":"https://pith.science/api/pith-number/ZQCN7VDR4THIM2X2LQH3IENOJM/graph.json","fetch_events":"https://pith.science/api/pith-number/ZQCN7VDR4THIM2X2LQH3IENOJM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZQCN7VDR4THIM2X2LQH3IENOJM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZQCN7VDR4THIM2X2LQH3IENOJM/action/storage_attestation","attest_author":"https://pith.science/pith/ZQCN7VDR4THIM2X2LQH3IENOJM/action/author_attestation","sign_citation":"https://pith.science/pith/ZQCN7VDR4THIM2X2LQH3IENOJM/action/citation_signature","submit_replication":"https://pith.science/pith/ZQCN7VDR4THIM2X2LQH3IENOJM/action/replication_record"}},"created_at":"2026-06-09T02:09:10.594471+00:00","updated_at":"2026-06-09T02:09:10.594471+00:00"}