{"paper":{"title":"Forcing Posets with Large Dimension to Contain Large Standard Examples","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Attila P\\'or, Csaba Bir\\'o, Peter Hamburger, William T. Trotter","submitted_at":"2014-02-20T19:44:12Z","abstract_excerpt":"The dimension of a poset $P$, denoted $\\dim(P)$, is the least positive integer $d$ for which $P$ is the intersection of $d$ linear extensions of $P$. The maximum dimension of a poset $P$ with $|P|\\le 2n+1$ is $n$, provided $n\\ge2$, and this inequality is tight when $P$ contains the standard example $S_n$. However, there are posets with large dimension that do not contain the standard example $S_2$. Moreover, for each fixed $d\\ge2$, if $P$ is a poset with $|P|\\le 2n+1$ and $P$ does not contain the standard example $S_d$, then $\\dim(P)=o(n)$. Also, for large $n$, there is a poset $P$ with $|P|=2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.5113","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"}