{"paper":{"title":"Linear $d$-polychromatic $Q_{d-1}$-colorings of the Hypercube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Offner, Eugene Han","submitted_at":"2017-12-07T05:25:56Z","abstract_excerpt":"Let $n \\ge d \\ge \\ell \\ge 1$ be integers, and denote the $n$-dimensional hypercube by $Q_n$. A coloring of the $\\ell$-dimensional subcubes $Q_\\ell$ in $Q_n$ is called a $Q_\\ell$-coloring. Such a coloring is $d$-polychromatic if every $Q_d$ in the $Q_n$ contains a $Q_\\ell$ of every color. In this paper we consider a specific class of $Q_\\ell$-colorings that are called linear. Given $\\ell$ and $d$, let $p_{lin}^\\ell(d)$ be the largest number of colors such that there is a $d$-polychromatic linear $Q_\\ell$-coloring of $Q_n$ for all $n \\ge d$. We prove that for all $d \\ge 3$, $p_{lin}^{d-1}(d) = 2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02496","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"}