{"paper":{"title":"A Common Generalization of the Theorems of Erd\\H{o}s-Ko-Rado and Hilton-Milner","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bor-Liang Chen, Ko-Wei Lih, Kuo-Ching Huang, Wei-Tian Li","submitted_at":"2013-04-06T06:44:15Z","abstract_excerpt":"Let $m$, $n$, and $k$ be integers satisfying $0 < k \\leq n < 2k \\leq m$. A family of sets $\\mathcal{F}$ is called an $(m,n,k)$-intersecting family if $\\binom{[n]}{k} \\subseteq \\mathcal{F} \\subseteq \\binom{[m]}{k}$ and any pair of members of $\\mathcal{F}$ have nonempty intersection. Maximum $(m,k,k)$- and $(m,k+1,k)$-intersecting families are determined by the theorems of Erd\\H{o}s-Ko-Rado and Hilton-Milner, respectively. We determine the maximum families for the cases $n = 2k-1, 2k-2, 2k-3$, and $m$ sufficiently large."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1861","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"}