{"paper":{"title":"Strong non-principality of positive codegree Tur\\'an density","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jun Gao, Levente Bodn\\'ar, Mingyuan Rong, Oleg Pikhurko, Shumin Sun","submitted_at":"2026-06-18T17:09:42Z","abstract_excerpt":"The \\emph{minimum positive codegree} $\\delta^+_{k-1}(G)$ of a $k$-graph $G$ is the minimum, over all $(k-1)$-sets that lie in at least one edge, of the number of edges containing that set. The \\emph{positive codegree Tur\\'an density} of a $k$-graph family $\\mathcal{F}$ is the asymptotically maximum value of $\\delta^+_{k-1}(G)/n$ over all $\\mathcal{F}$-free $k$-graphs $G$ with $n\\to\\infty$ vertices. In this note, we establish a strong version of non-principality with respect to this density by proving that for every $k\\ge3$ there exist two $k$-graphs $F_1$ and $F_2$ such that\n  $$\n  0<\\gamma^+("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.20494","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/2606.20494/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"}