{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:X6HVUY2FNCNANWGAUM2Y5S56FO","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"867a7e227c5f79420706e4091c7cb20714cd5d9be44cfcb5f321e746348bee4c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-02T04:12:00Z","title_canon_sha256":"38abe6486861ecd3c3951e97056195a6d169b6cd53a8d60b413ea25295f60281"},"schema_version":"1.0","source":{"id":"2607.01680","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01680","created_at":"2026-07-03T01:17:26Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01680v1","created_at":"2026-07-03T01:17:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01680","created_at":"2026-07-03T01:17:26Z"},{"alias_kind":"pith_short_12","alias_value":"X6HVUY2FNCNA","created_at":"2026-07-03T01:17:26Z"},{"alias_kind":"pith_short_16","alias_value":"X6HVUY2FNCNANWGA","created_at":"2026-07-03T01:17:26Z"},{"alias_kind":"pith_short_8","alias_value":"X6HVUY2F","created_at":"2026-07-03T01:17:26Z"}],"graph_snapshots":[{"event_id":"sha256:2b6c9d84abc161fc42a67309b062d5c01730aeb25c74b35c1d23788c794a1292","target":"graph","created_at":"2026-07-03T01:17:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.01680/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For graphs $F$ and $H$, let $\\mathrm{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Very recently, Janzer, Longbrake, and Yepremyan proved that for $3<a\\leq b$ and sufficiently large $t$, \\begin{equation*} \\mathrm{ex}(n,K_{a,b},K_{3,t})=\\Theta_{a,b,t}(n^3). \\end{equation*} Later, Hou, Hu, and Wang made this threshold explicit by showing that the conclusion holds for all $t\\geq 2\\max\\{3,\\lceil b/2\\rceil\\}+1$. In particular, for every even $b\\geq 6$, this matches the necessary threshold $t=b+1$. In this paper, we resolve the remaining case where $b$ is od","authors_text":"Jing Wang, Junpeng Zhou, Zixuan Yang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-02T04:12:00Z","title":"On the generalized Tur\\'{a}n number of the complete bipartite graph $K_{3,b+1}$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01680","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:e1ad714a4ceac59400b57e545e1e1ff09ec19ea52ffbbac41b09d45153e0e917","target":"record","created_at":"2026-07-03T01:17:26Z","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":"867a7e227c5f79420706e4091c7cb20714cd5d9be44cfcb5f321e746348bee4c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-02T04:12:00Z","title_canon_sha256":"38abe6486861ecd3c3951e97056195a6d169b6cd53a8d60b413ea25295f60281"},"schema_version":"1.0","source":{"id":"2607.01680","kind":"arxiv","version":1}},"canonical_sha256":"bf8f5a6345689a06d8c0a3358ecbbe2b951efbade27228e63671a372df7e4734","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf8f5a6345689a06d8c0a3358ecbbe2b951efbade27228e63671a372df7e4734","first_computed_at":"2026-07-03T01:17:26.595814Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T01:17:26.595814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZstbpV6cwQv4Z5xg/m9VqABsHkln8xf6aEBrO0BoPm+tZlPiRWBcQyUj8F2fHwYJkbGyVivHOVovEbHSoee3DA==","signature_status":"signed_v1","signed_at":"2026-07-03T01:17:26.596277Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.01680","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e1ad714a4ceac59400b57e545e1e1ff09ec19ea52ffbbac41b09d45153e0e917","sha256:2b6c9d84abc161fc42a67309b062d5c01730aeb25c74b35c1d23788c794a1292"],"state_sha256":"c3df4f22afab87716afa7e8d6ed2c891eef7667a3dc5b815f92327c18979ffe2"}