{"paper":{"title":"Bipartite-ness under smooth conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Ma, Sean Longbrake, Tao Jiang","submitted_at":"2021-09-03T04:58:13Z","abstract_excerpt":"Given a family $\\mathcal{F}$ of bipartite graphs, the {\\it Zarankiewicz number} $z(m,n,\\mathcal{F})$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does not contain any member of $\\mathcal{F}$ as a subgraph (such $G$ is called {\\it $\\mathcal{F}$-free}). For $1\\leq \\beta<\\alpha<2$, a family $\\mathcal{F}$ of bipartite graphs is $(\\alpha,\\beta)$-{\\it smooth} if for some $\\rho>0$ and every $m\\leq n$, $z(m,n,\\mathcal{F})=\\rho m n^{\\alpha-1}+O(n^\\beta)$. Motivated by their work on a conjecture of Erd\\H{o}s and Simonovits on compactness and a classic result of Andr\\'asfai, E"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.01311","kind":"arxiv","version":4},"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/2109.01311/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"}