{"paper":{"title":"Cross-Sperner families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bal\\'azs Patk\\'os, Cory Palmer, D\\'aniel Gerbner, Nathan Lemons, Vajk Sz\\'ecsi","submitted_at":"2011-04-20T10:45:18Z","abstract_excerpt":"A pair of families $(\\cF,\\cG)$ is said to be \\emph{cross-Sperner} if there exists no pair of sets $F \\in \\cF, G \\in \\cG$ with $F \\subseteq G$ or $G \\subseteq F$. There are two ways to measure the size of the pair $(\\cF,\\cG)$: with the sum $|\\cF|+|\\cG|$ or with the product $|\\cF|\\cdot |\\cG|$. We show that if $\\cF, \\cG \\subseteq 2^{[n]}$, then $|\\cF||\\cG| \\le 2^{2n-4}$ and $|\\cF|+|\\cG|$ is maximal if $\\cF$ or $\\cG$ consists of exactly one set of size $\\lceil n/2 \\rceil$ provided the size of the ground set $n$ is large enough and both $\\cF$ and $\\cG$ are non-empty."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3988","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"}