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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 $H$.","authors_text":"Liana Yepremyan, Oliver Janzer, Sean Longbrake","cross_cats":["cs.DM"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-08T17:53:07Z","title":"On the generalized Tur\\'an number of complete bipartite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.09801","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:09d06728cdb4b890e6100a313c8cbbc47b4e58a8a42b9c1574767de61d7e26f5","target":"record","created_at":"2026-06-09T02:09:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4082da0b3408dfce2db433ec3a74257dfe4f123962a3b0fce438facaa9099e19","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-08T17:53:07Z","title_canon_sha256":"7108c969af300ef88e49e848d61347949fd8db247fd198c358425b5925c72a69"},"schema_version":"1.0","source":{"id":"2606.09801","kind":"arxiv","version":1}},"canonical_sha256":"cc04dfd471e4ce866afa5c0fb411ae4b1676ceb9a2e03c7081d8d2c94e59ebb8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cc04dfd471e4ce866afa5c0fb411ae4b1676ceb9a2e03c7081d8d2c94e59ebb8","first_computed_at":"2026-06-09T02:09:10.594362Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T02:09:10.594362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"W8tq5CWFSWev4urvEblDrC4wUppNAHZZE2N8o/yAIMm97ta9pjffoD4/if4FC342KkWgSnqMbb/lhKfXxnryBg==","signature_status":"signed_v1","signed_at":"2026-06-09T02:09:10.595070Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.09801","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:09d06728cdb4b890e6100a313c8cbbc47b4e58a8a42b9c1574767de61d7e26f5","sha256:82adf79b4c95a96a4651a3be3a9255fbf396af5c62802e222a7d52e7d08f3094"],"state_sha256":"5c636aab18e682b385cd75f5c407521d3d4c1b9ff0e29309b87da29953d702c0"}