{"paper":{"title":"Vanishing orders, suspensions and zero degree Tur\\'an densities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Haotian Yang, Hong Liu, Jiangdong Ai, Laihao Ding","submitted_at":"2026-03-06T07:13:04Z","abstract_excerpt":"For integers $1\\le \\ell<k$, the $\\ell$-degree Tur\\'an density $\\pi_\\ell(F)$ measures the minimum $\\ell$-degree threshold that forces a copy of a fixed $k$-uniform hypergraph $F$, generalizing both the classical Tur\\'an density $\\pi_1$ and the codegree Tur\\'an density $\\pi_{k-1}$. Motivated by Erd\\H{o}s' characterization of $k$-graphs with zero Tur\\'an density, we study the structural implications of vanishing $\\ell$-degree Tur\\'an density.\n  Our main result concerns the case $\\ell=2$. We prove that, for every $k\\ge3$, if a $k$-graph $F$ satisfies $\\pi_2(F)=0$, then $F$ admits a $2$-vanishing o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.05973","kind":"arxiv","version":2},"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/2603.05973/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"}