{"paper":{"title":"The minimum number of vertices in uniform hypergraphs with given domination number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bal\\'azs Patk\\'os, Csilla Bujt\\'as, M\\'at\\'e Vizer, Zsolt Tuza","submitted_at":"2016-03-11T08:17:06Z","abstract_excerpt":"The \\textit{domination number} $\\gamma(\\mathcal{H})$ of a hypergraph $\\mathcal{H}=(V(\\mathcal{H}),E(\\mathcal{H})$ is the minimum size of a subset $D\\subset V(\\mathcal{H}$ of the vertices such that for every $v\\in V(\\mathcal{H})\\setminus D$ there exist a vertex $d \\in D$ and an edge $H\\in E(\\mathcal{H})$ with $v,d\\in H$. We address the problem of finding the minimum number $n(k,\\gamma)$ of vertices that a $k$-uniform hypergraph $\\mathcal{H}$ can have if $\\gamma(\\mathcal{H})\\ge \\gamma$ and $\\mathcal{H}$ does not contain isolated vertices. We prove that $$n(k,\\gamma)=k+\\Theta(k^{1-1/\\gamma})$$ an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03557","kind":"arxiv","version":2},"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"}