{"paper":{"title":"Lower bounds on the sizes of defining sets in full $n$-Latin squares and full designs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nicholas Cavenagh","submitted_at":"2017-08-21T02:10:07Z","abstract_excerpt":"The full $n$-Latin square is the $n\\times n$ array with symbols $1,2,\\dots ,n$ in each cell. In this paper we show, as part of a more general result, that any defining set for the full $n$-Latin square has size $n^3(1-o(1))$. The full design $N(v,k)$ is the unique simple design with parameters $(v,k,{v-2 \\choose k-2})$; that is, the design consisting of all subsets of size $k$ from a set of size $v$. We show that any defining set for the full design $N(v,k)$ has size ${v\\choose k}(1-o(1))$ (as $v-k$ becomes large). These results improve existing results and are asymptotically optimal. In parti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.06058","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"}