{"paper":{"title":"Boolean algebras and Lubell functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kevin G. Milans, Linyuan Lu, Travis Johnston","submitted_at":"2013-07-12T03:21:44Z","abstract_excerpt":"Let $2^{[n]}$ denote the power set of $[n]:=\\{1,2,..., n\\}$. A collection $\\B\\subset 2^{[n]}$ forms a $d$-dimensional {\\em Boolean algebra} if there exist pairwise disjoint sets $X_0, X_1,..., X_d \\subseteq [n]$, all non-empty with perhaps the exception of $X_0$, so that $\\B={X_0\\cup \\bigcup_{i\\in I} X_i\\colon I\\subseteq [d]}$. Let $b(n,d)$ be the maximum cardinality of a family $\\F\\subset 2^X$ that does not contain a $d$-dimensional Boolean algebra. Gunderson, R\\\"odl, and Sidorenko proved that $b(n,d) \\leq c_d n^{-1/2^d} \\cdot 2^n$ where $c_d= 10^d 2^{-2^{1-d}}d^{d-2^{-d}}$.\n  In this paper, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.3312","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"}