{"paper":{"title":"Explicit thresholds in a generalized Tur\\'an problem for \\(K_{3,t}\\)-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Caiyun Hu, Hezhi Wang, Jianfeng Hou","submitted_at":"2026-06-17T15:54:48Z","abstract_excerpt":"For graphs $F$ and $H$, let $\\ex(n,F,H)$ denote the maximum number of copies of $F$ in an $n$-vertex $H$-free graph. Janzer, Longbrake and Yepremyan recently proved that, for fixed $3<a\\le b$ and sufficiently large $t$, \\[\n  \\ex(n,K_{a,b},K_{3,t})=\\Theta(n^3). \\] We make their threshold explicit, showing that this conclusion holds for all $t\\ge \\tau(b):=2\\max\\{3,\\lceil b/2\\rceil\\}+1.$ In particular, for every even $b\\ge 6$, this matches the necessary threshold $t=b+1$. The main new ingredient is an explicit finite-field point set whose plane sections are controlled directly, rather than throug"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.19217","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/2606.19217/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"}