{"paper":{"title":"A note on traces of set families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Balazs Patkos","submitted_at":"2011-11-20T14:23:31Z","abstract_excerpt":"A family of sets $\\mathcal{F} \\subseteq 2^{[n]}$ is defined to be $l$-trace $k$-Sperner if for any $l$-subset $L$ of $[n]$ the family of traces $\\mathcal{F}|_L=\\{F \\cap L: F \\in \\mathcal{F}\\}$ does not contain any chain of length $k+1$. In this paper we prove that for any positive integers $l',k$ with $l'<k$ if $\\mathcal{F}$ is $(n-l')$-trace $k$-Sperner, then $|\\mathcal{F}| \\le (k-l'+o(1))\\binom{n}{\\lfloor n/2\\rfloor}$ and this bound is asymptotically tight."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.4636","kind":"arxiv","version":3},"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"}