{"paper":{"title":"An Analogue of Hilton-Milner Theorem for Set Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cheng Yeaw Ku, Kok Bin Wong","submitted_at":"2011-09-02T11:59:25Z","abstract_excerpt":"Let $\\mathcal{B}(n)$ denote the collection of all set partitions of $[n]$. Suppose $\\mathcal{A} \\subseteq \\mathcal{B}(n)$ is a non-trivial $t$-intersecting family of set partitions i.e. any two members of $\\A$ have at least $t$ blocks in common, but there is no fixed $t$ blocks of size one which belong to all of them. It is proved that for sufficiently large $n$ depending on $t$, \\[ |\\mathcal{A}| \\le B_{n-t}-\\tilde{B}_{n-t}-\\tilde{B}_{n-t-1}+t \\] where $B_{n}$ is the $n$-th Bell number and $\\tilde{B}_{n}$ is the number of set partitions of $[n]$ without blocks of size one. Moreover, equality h"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.0417","kind":"arxiv","version":1},"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"}