{"paper":{"title":"Some exact results on $4$-cycles: stability and supersaturation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jialin He, Jie Ma, Tianchi Yang","submitted_at":"2019-12-02T18:24:46Z","abstract_excerpt":"Extremal problems on the $4$-cycle $C_4$ played a heuristic important role in the development of extremal graph theory. A fundamental theorem of F\\\"uredi states that the Tur\\'an number $ex(q^2+q+1, C_4)\\leq \\frac12 q(q+1)^2$ holds for every $q\\geq 14$, which matches with the classic construction of Erd\\H{o}s-R{\\'e}nyi-S\\'os and Brown from finite geometry for prime powers $q$.\n  Very recently, we obtained the first stability result on F\\\"uredi's theorem, by showing that for large even $q$, every $(q^2+q+1)$-vertex $C_4$-free graph with more than $\\frac12 q(q+1)^2-0.2q$ edges must be a spanning "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1912.00986","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/1912.00986/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"}