{"paper":{"title":"On the generalized Tur\\'{a}n number of the complete bipartite graph $K_{3,b+1}$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jing Wang, Junpeng Zhou, Zixuan Yang","submitted_at":"2026-07-02T04:12:00Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01680","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/2607.01680/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"}