{"paper":{"title":"On the distances between Latin squares and the smallest defining set size","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nicholas Cavenagh, Reshma Ramadurai","submitted_at":"2016-02-24T22:25:17Z","abstract_excerpt":"In this note we show that for each Latin square $L$ of order $n\\geq 2$, there exists a Latin square $L'\\neq L$ of order $n$ such that $L$ and $L'$ differ in at most $8\\sqrt{n}$ cells. Equivalently, each Latin square of order $n$ contains a Latin trade of size at most $8\\sqrt{n}$. We also show that the size of the smallest defining set in a Latin square is $\\Omega(n^{3/2})$. %That is, there are constants $c$ and $n_0$ such that for any $n>n_0$ the size of the smallest defining %set of order $n$ is at least $cn^{3/2}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.07734","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"}