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We study the problem to find the smallest $r$ such that there is a family $\\mathcal{A}$ of 2-subsets and 3-subsets of $[n]=\\{1,2,...,n\\}$ with the following properties: (1) $\\mathcal{A}$ is an antichain, i.e. no member of $\\mathcal A$ is a subset of any other member of $\\mathcal A$, (2) $\\mathcal A$ is maximal, i.e. for every $X\\in 2^{[n]}\\setminus\\mathcal A$ there is an $A\\in\\mathcal A$ with $X\\subseteq A$ or $A\\subseteq X$, and (3) $\\mathcal A$ is $r$-regular, i.e. every point $x\\in[n]$ is contained in exactly $r$ members of $\\mathcal A$. We prove lowe"},"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":"1206.3752","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-17T12:49:39Z","cross_cats_sorted":[],"title_canon_sha256":"da9b87c4c3881abd257f4b5daa3ee12466ee811953f53b904a3b16aa10c66b81","abstract_canon_sha256":"61fe52e0854003bc5afb203d4ba9c3267f3ad82610804066221fdd254dccb38b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:31.020268Z","signature_b64":"kl5GjXrBSdRxcZEV/vXCc8rmVzuODz7N7Evc6unl6+WedWMnwyCb2pkDhYSieOP1Nvex68GNMt9MNxVKV5/9AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"703a136c4d3e0b62bed8a13ba038256b2469146a65633ed4137eefee21f85f1d","last_reissued_at":"2026-05-18T01:24:31.019702Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:31.019702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimizing the regularity of maximal regular antichains of 2- and 3-sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Reiher, Ian T. Roberts, Thomas Kalinowski, Uwe Leck","submitted_at":"2012-06-17T12:49:39Z","abstract_excerpt":"Let $n\\geqslant 3$ be a natural number. We study the problem to find the smallest $r$ such that there is a family $\\mathcal{A}$ of 2-subsets and 3-subsets of $[n]=\\{1,2,...,n\\}$ with the following properties: (1) $\\mathcal{A}$ is an antichain, i.e. no member of $\\mathcal A$ is a subset of any other member of $\\mathcal A$, (2) $\\mathcal A$ is maximal, i.e. for every $X\\in 2^{[n]}\\setminus\\mathcal A$ there is an $A\\in\\mathcal A$ with $X\\subseteq A$ or $A\\subseteq X$, and (3) $\\mathcal A$ is $r$-regular, i.e. every point $x\\in[n]$ is contained in exactly $r$ members of $\\mathcal A$. 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