{"paper":{"title":"Most binary matrices have no small defining set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anita Liebenau, Carly Bodkin, Ian M. Wanless","submitted_at":"2019-08-04T03:43:07Z","abstract_excerpt":"Consider a matrix $M$ chosen uniformly at random from a class of $m \\times n$ matrices of zeros and ones with prescribed row and column sums. A partially filled matrix $D$ is a $\\mathit{defining}$ $\\mathit{set}$ for $M$ if $M$ is the unique member of its class that contains the entries in $D$. The $\\mathit{size}$ of a defining set is the number of filled entries. A $\\mathit{critical}$ $\\mathit{set}$ is a defining set for which the removal of any entry stops it being a defining set. For some small fixed $\\epsilon>0$, we assume that $n\\le m=o(n^{1+\\epsilon})$, and that $\\lambda\\le1/2$, where $\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01267","kind":"arxiv","version":2},"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/1908.01267/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"}