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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},"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1812.08282","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-19T22:56:42Z","cross_cats_sorted":[],"title_canon_sha256":"d893716d487a0c807923a4a74fc41d8a4f684ef597290b5bd6a6d27b9eca3869","abstract_canon_sha256":"650912daff864da6e0dc077a8190d4525c32a8ab8d2702f05cb7abdcbb18b1d7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:51.739621Z","signature_b64":"htUWqEofDeh5Aw6qi8iTvpTf/oQtLLqvLcmGamZZmL8TRh9nxnxsUoSXFwCR9k1qS22ji+/6K+xMNuSrUcYKCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aea5f621803732353663770a8494dbf9b37fc250d8e1b003dab7b159e1cc8f8f","last_reissued_at":"2026-05-17T23:57:51.738928Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:51.738928Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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