{"paper":{"title":"Latin Cubes with Forbidden Entries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Carl Johan Casselgren, Klas Markstr\\\"om, Lan Anh Pham","submitted_at":"2018-09-07T10:31:55Z","abstract_excerpt":"We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant $\\gamma > 0$ such that if $n=2^k$ and $A$ is $3$-dimensional $n\\times n\\times n$ array where every cell contains at most $\\gamma n$ symbols, and every symbol occurs at most $\\gamma n$ times in every line of $A$, then $A$ is {\\em avoidable}; that is, there is a Latin cube $L$ of order $n$ such that for every $1\\leq i,j,k\\leq n$, the symbol in position $(i,j,k)$ of $L$ does not appear in the corresponding cell of $A$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.02392","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"}