{"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 $\\mathcal A$. We prove lowe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.3752","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}