{"paper":{"title":"The maximum, spectrum and supremum for critical set sizes in (0,1)-matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Liam K. Wright, Nicholas J. Cavenagh","submitted_at":"2018-12-19T22:56:42Z","abstract_excerpt":"If $D$ is a partially filled-in $(0,1)$-matrix with a unique completion to a $(0,1)$-matrix $M$ (with prescribed row and column sums), we say that $D$ is a {\\em defining set} for $M$. A {\\em critical set} is a minimal defining set (the deletion of any entry results in more than one completion). We give a new classification of critical sets in $(0,1)$-matrices and apply this theory to $\\Lambda_{2m}^m$, the set of $(0,1)$-matrices of dimensions $2m\\times 2m$ with uniform row and column sum $m$.\n  The smallest possible size for a defining set of a matrix in $\\Lambda_{2m}^m$ is $m^2$\n  \\cite{Cav},"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.08282","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"}