{"paper":{"title":"A random version of Sperner's theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Treglown, J\\'ozsef Balogh, Richard Mycroft","submitted_at":"2014-04-20T23:10:59Z","abstract_excerpt":"Let $\\mathcal{P}(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\\mathcal{P}(n,p)$ be obtained from $\\mathcal{P}(n)$ by selecting elements from $\\mathcal{P}(n)$ independently at random with probability $p$. A classical result of Sperner asserts that every antichain in $\\mathcal{P}(n)$ has size at most that of the middle layer, $\\binom{n}{\\lfloor n/2 \\rfloor}$. In this note we prove an analogous result for $\\mathcal{P} (n,p)$: If $pn \\rightarrow \\infty$ then, with high probability, the size of the largest antichain in $\\mathcal{P}(n,p)$ is at most $(1+o(1)) p \\binom{n}{\\lfloor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.5079","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":""},"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"}