{"paper":{"title":"Random Tur\\'an Theorem for the Fano Plane","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Ilay Hoshen","submitted_at":"2026-07-30T11:45:43Z","abstract_excerpt":"Let $F$ denote the Fano plane, the $3$-uniform hypergraph with $7$ vertices and $7$ edges. Frankl and F\\\"uredi, and independently Keevash and Sudakov, proved that the largest $F$-free subhypergraph of $K_n^{(3)}$ is bipartite. In this paper, we determine the sharp threshold for this property in the random setting. We show that for $\\hat{p} = \\Theta_F \\cdot n^{-2/3} \\left(\\log n\\right)^{1/6}$, where $\\Theta_F$ is an explicit constant depending on $F$, we have: (i) if $(1+\\epsilon) \\hat{p} \\le p = o(1)$, then with high probability every largest $F$-free subhypergraph of $G_{n,p}^{(3)}$ is bipart"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28071","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/2607.28071/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"}