{"paper":{"title":"Two-part set systems","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, Dhruv Mubayi, Nathan Lemons, P\\'eter L. Erd\\H{o}s","submitted_at":"2011-10-01T13:41:52Z","abstract_excerpt":"The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \\cup X_2$, and $\\cF$ is a family of subsets of $X$ such that no two sets $A, B \\in \\cF$ satisfy $A \\subset B$ (or $B \\subset A$) and $A \\cap X_i=B \\cap X_i$ for some $i$, then $|\\cF| \\le {n \\choose \\lfloor n/2 \\rfloor}$. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfiy various combinations of these properties on one or both parts $X_1$, $X_2$. Along the way, we prove the following new resul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.0099","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"}