{"paper":{"title":"Set families with forbidden subposets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kevin G. Milans, Linyuan Lu","submitted_at":"2014-08-04T11:22:25Z","abstract_excerpt":"Let $F$ be a family of subsets of $\\{1,\\ldots,n\\}$. We say that $F$ is $P$-free if the inclusion order on $F$ does not contain $P$ as an induced subposet. The \\emph{Tur\\'an function} of $P$, denoted $\\pi^*(n,P)$, is the maximum size of a $P$-free family of subsets of $\\{1,\\ldots,n\\}$. We show that $\\pi^*(n,P) \\le (4r + O(\\sqrt{r}))\\binom{n}{n/2}$ if $P$ is an $r$-element poset of height at most $2$. We also show that $\\pi^*(n,S_r) = (r+O(\\sqrt{r}))\\binom{n}{n/2}$ where $S_r$ is the standard example on $2r$ elements, and that $\\pi^*(n,B_2) \\le (2.583+o(1))\\binom{n}{n/2}$, where $B_2$ is the $2$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.0646","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"}