{"paper":{"title":"A note on the size of N-free families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ryan R. Martin, Shanise Walker","submitted_at":"2016-09-07T04:40:54Z","abstract_excerpt":"The $\\mathcal{N}$ poset consists of four distinct sets $W,X,Y,Z$ such that $W\\subset X$, $Y\\subset X$, and $Y\\subset Z$ where $W$ is not necessarily a subset of $Z$. A family $\\mathcal{F}$ as a subposet of the $n$-dimensional Boolean lattice, $\\mathcal{B}_n$, is $\\mathcal{N}$-free if it does not contain $\\mathcal{N}$ as a subposet. Let $\\text{La}(n, \\mathcal{N})$ be the size of a largest $\\mathcal{N}$-free family in $\\mathcal{B}_n$. Katona and Tarj\\'{a}n proved that $\\text{La}(n,\\mathcal{N})\\geq {n \\choose k}+A(n,4,k+1)$, where $k=\\lfloor n/2\\rfloor$ and $A(n, 4, k+1)$ is the size of a single-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.02442","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"}