{"paper":{"title":"Exact Tur\\'{a}n number of the Fano plane in the $\\ell_2$-norm","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jianfeng Hou, Xizhi Liu, Yixiao Zhang","submitted_at":"2025-07-16T15:52:49Z","abstract_excerpt":"A classical object in hypergraph Tur\\'{a}n theory is the Fano plane $\\mathbb{F}$, the unique linear $3$-graph on seven vertices with seven edges. The Tur\\'{a}n density and exact Tur\\'{a}n number of $\\mathbb{F}$, first proposed as a problem by S\\'{o}s \\cite{Sos76} in the 1970s, were determined through a sequence of works by De Caen-F\\\"{u}redi \\cite{DCF00}, F\\\"{u}redi-Simonovits \\cite{FS05}, Keevash-Sudakov \\cite{KS05}, and Bellmann-Reiher \\cite{BR19}. Addressing a conjecture of Balogh-Clemen-Lidick\\'{y} \\cite[Conjecture 3.1]{BCL22a}, we establish an Andr\\'{a}sfai-Erd\\H{o}s-S\\'{o}s-type stabilit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.12354","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/2507.12354/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"}