{"paper":{"title":"A Simple Proof Characterizing Interval Orders with Interval Lengths between 1 and $k$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ann Trenk, Garth Isaak, Simona Boyadzhiyska","submitted_at":"2017-09-01T13:53:49Z","abstract_excerpt":"A poset $P= (X, \\prec)$ has an interval representation if each $x \\in X$ can be assigned a real interval $I_x$ so that $x \\prec y$ in $P$ if and only if $I_x$ lies completely to the left of $I_y$. Such orders are called \\emph{interval orders}. Fishburn proved that for any positive integer $k$, an interval order has a representation in which all interval lengths are between $1$ and $k$ if and only if the order does not contain $\\mathbf{(k+2)+1}$ as an induced poset. In this paper, we give a simple proof of this result using a digraph model."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00313","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"}