{"paper":{"title":"On the size of the minimum critical set of a Latin square","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ebadollah S. Mahmoodian, Hamed Hatami, Mahya Ghandehari","submitted_at":"2006-12-31T04:36:41Z","abstract_excerpt":"A critical set in an $n \\times n$ array is a set $C$ of given entries, such that there exists a unique extension of $C$ to an $n\\times n$ Latin square and no proper subset of $C$ has this property. For a Latin square $L$, $\\scs{L}$ denotes the size of the smallest critical set of $L$, and $\\scs{n}$ is the minimum of $\\scs{L}$ over all Latin squares $L$ of order $n$. We find an upper bound for the number of partial Latin squares of size $k$ and prove that $$n^2-(e+o(1))n^{10/6} \\le \\max \\scs{L} \\le n^2-\\frac{\\sqrt{\\pi}}{2}n^{9/6}.$$ % This improves a result of N. Cavenagh (Ph.D. thesis, The Uni"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0701015","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0701015/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}