{"paper":{"title":"$Q_2$-free families in the Boolean lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jacob Manske, Maria Axenovich, Ryan R. Martin","submitted_at":"2009-12-26T20:27:08Z","abstract_excerpt":"For a family $\\mathcal{F}$ of subsets of [n]=\\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that $\\mathcal{F}$ is P-free if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the largest size of a P-free family of subsets of [n]. Let $Q_2$ be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that $2N -o(N) \\leq ex(n, Q_2)\\leq 2.283261N +o(N), $ where $N = \\binom{n}{\\lfloor n/2 \\rfloor}$. We also prove that the largest $Q_2$-free family of subsets of [n] having at most three different sizes has at "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.5039","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"}