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First, for n=T-c with 1<=c<=sT/(s+2), Roman's bound equals the elementary counting bound (s+1)(T-c)+floor(2c/s), adding nothing beyond a budget inequality and convexity. Second, attainment forces all but at most one column to have size s+1 or s+2; each such column has a point lying in a nu"},"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":"2608.07607","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-08-06T20:20:47Z","cross_cats_sorted":[],"title_canon_sha256":"1846f9e0c3143a6eae8f1380522566eb48ba3765f15b4778d082482146592d78","abstract_canon_sha256":"43fbf43880a88c4604539074e5b0f771dd90c0bea4933f91de9185408476d4c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-11T00:15:28.777767Z","signature_b64":"ul0z2IfESnxbtuqrLBHEEMUrZ4kO916/URn7juXoEm2AOyRdnN0dOWW/puSV8pkEF5Oz5TEVN5mR9+nCqXj8Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cdf2e14e662cbc77f3c11a16d453b9a552eaf57a677b80fd6ccebc9ec37e64d6","last_reissued_at":"2026-08-11T00:15:28.775548Z","signature_status":"signed_v1","first_computed_at":"2026-08-11T00:15:28.775548Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A congruence obstruction to Roman's bound for Zarankiewicz numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ankan Sadhu","submitted_at":"2026-08-06T20:20:47Z","abstract_excerpt":"Let z(m,n;s,t) be the largest number of ones in an m x n zero-one matrix with no s x t all-ones submatrix. 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