{"paper":{"title":"Families that remain $k$-Sperner even after omitting an element of their ground set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Balazs Patkos","submitted_at":"2012-07-12T11:12:21Z","abstract_excerpt":"A family $\\cF\\subseteq 2^{[n]}$ of sets is said to be $l$-trace $k$-Sperner if for any $l$-subset $L \\subset [n]$ the family $\\cF|_L=\\{F|_L:F \\in \\cF\\}=\\{F \\cap L: F \\in \\cF\\}$ is $k$-Sperner, i.e. does not contain any chain of length $k+1$. The maximum size that an $l$-trace $k$-Sperner family $\\cF \\subseteq 2^{[n]}$ can have is denoted by $f(n,k,l)$. For pairs of integers $l<k$, if in a family $\\cG$ every pair of sets satisfies $||G_1|-|G_2||<k-l$, then $\\cG$ possesses the $(n-l)$-trace $k$-Sperner property. Among such families, the largest one is $\\cF_0=\\{F\\in 2^{[n]}: \\lfloor \\frac{n-(k-l)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2923","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":""},"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"}