{"paper":{"title":"Tur\\'{a}n density of tight cycles minus one edge in the $\\ell_2$-norm","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hongbin Zhao, Jianfeng Hou, Jinghua Deng, Levente Bodn\\'ar, Xizhi Liu","submitted_at":"2025-07-01T14:43:42Z","abstract_excerpt":"The $3$-uniform tight $\\ell$-cycle minus one edge $C_{\\ell}^{3-}$ is the $3$-graph on $\\ell$ vertices consisting of $\\ell-1$ consecutive triples in the cyclic order. We show that for every integer $\\ell \\ge 5$ satisfying $\\ell\\not\\equiv 0\\pmod3$, every $C_{\\ell}^{3-}$-free $3$-graph whose $\\ell_2$-norm, that is, the sum of codegree squares, is close to the maximum must be structurally close to the iterative blowup of a single triple. This confirms a conjecture of Balogh--Clemen--Lidick\\'{y}~[Surveys in combinatorics 2022, 21-63] in a stronger form."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.00812","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/2507.00812/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"}