{"paper":{"title":"Uniformity thresholds for the asymptotic size of extremal Berge-$F$-free hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abhishek Methuku, Casey Tompkins, D\\'aniel Gr\\'osz","submitted_at":"2018-03-05T22:41:00Z","abstract_excerpt":"Let $F = (U,E)$ be a graph and $\\mathcal{H} = (V,\\mathcal{E})$ be a hypergraph. We say that $\\mathcal{H}$ contains a Berge-$F$ if there exist injections $\\psi:U\\to V$ and $\\varphi:E\\to \\mathcal{E}$ such that for every $e=\\{u,v\\}\\in E$, $\\{\\psi(u),\\psi(v)\\}\\subset\\varphi(e)$. Let $ex_r(n,F)$ denote the maximum number of hyperedges in an $r$-uniform hypergraph on $n$ vertices which does not contain a Berge-$F$.\n  For small enough $r$ and non-bipartite $F$, $ex_r(n,F)=\\Omega(n^2)$; we show that for sufficiently large $r$, $ex_r(n,F)=o(n^2)$. Let $thres(F) = \\min\\{r_0 :ex_r(n,F) = o(n^2) \\text{ fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.01953","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":""},"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"}