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Alm","submitted_at":"2012-06-06T02:17:23Z","abstract_excerpt":"For positive integers $m$ and $n$, define $f(m,n)$ to be the smallest integer such that any subset $A$ of the $m \\times n$ integer grid with $|A| \\geq f(m,n)$ contains a rectangle; that is, there are $x\\in [m]$ and $y \\in [n]$ and $d_{1},d_{2} \\in \\mathbb{Z}^{+}$ such that all four points $(x,y)$, $(x+d_{1},y)$, $(x,y+d_{2})$, and $(x+d_{1},y+d_{2})$ are contained in $A$. In \\cite{kovarisosturan}, K\\\"ovari, S\\'os, and Tur\\'an showed that $\\dlim_{k \\to \\infty}\\dfrac{f(k,k)}{k^{3/2}} = 1$. They also showed that whenever $p$ is a prime number, $f(p^{2},p^{2}+p) = p^{2}(p+1)+1$. We recover their a"},"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":"1206.1107","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-06T02:17:23Z","cross_cats_sorted":[],"title_canon_sha256":"4b7eaae104bda785fb4d44ae0244888015787c79b912537b3da134b7700258cc","abstract_canon_sha256":"fa9efc050cbd8e418e97422a73ce18e2e4209bc456b22916d733c17f5a9d1566"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:47.691348Z","signature_b64":"tvRCdSf+STnzo42g5U+RNzXb/N0yEMeZCG2A8iPOiZevONLn/x/goOHg+yrOPSnWUhEgV4RVE/NwSQshixm1Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1aa280ec611d994eafe042caa4b67f0c27bf3447c56d140afebed48775a302c0","last_reissued_at":"2026-05-18T03:08:47.690663Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:47.690663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A new approach to the results of K\\\"ovari, S\\'os, and Tur\\'an concerning rectangle-free subsets of the grid","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jacob Manske, Jeremy F. Alm","submitted_at":"2012-06-06T02:17:23Z","abstract_excerpt":"For positive integers $m$ and $n$, define $f(m,n)$ to be the smallest integer such that any subset $A$ of the $m \\times n$ integer grid with $|A| \\geq f(m,n)$ contains a rectangle; that is, there are $x\\in [m]$ and $y \\in [n]$ and $d_{1},d_{2} \\in \\mathbb{Z}^{+}$ such that all four points $(x,y)$, $(x+d_{1},y)$, $(x,y+d_{2})$, and $(x+d_{1},y+d_{2})$ are contained in $A$. In \\cite{kovarisosturan}, K\\\"ovari, S\\'os, and Tur\\'an showed that $\\dlim_{k \\to \\infty}\\dfrac{f(k,k)}{k^{3/2}} = 1$. They also showed that whenever $p$ is a prime number, $f(p^{2},p^{2}+p) = p^{2}(p+1)+1$. 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