{"paper":{"title":"On rainbow matchings for hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hongliang Lu, Xingxing Yu","submitted_at":"2016-11-06T07:29:29Z","abstract_excerpt":"For any posotive integer $m$, let $[m]:=\\{1,\\ldots,m\\}$. Let $n,k,t$ be positive integers. Aharoni and Howard conjectured that if, for $i\\in [t]$, $\\mathcal{F}_i\\subset[n]^k:= \\{(a_1,\\ldots,a_k): a_j\\in [n] \\mbox{ for } j\\in [k]\\}$ and $|\\mathcal{F}_i|>(t-1)n^{k-1}$, then there exist $M\\subseteq [n]^k$ such that $|M|=t$ and $|M\\cap \\mathcal{F}_i|=1$ for $i\\in [t]$ We show that this conjecture holds when $n\\geq 3(k-1)(t-1)$.\n  Let $n, t, k_1\\ge k_2\\geq \\ldots\\geq k_t $ be positive integers. Huang, Loh and Sudakov asked for the maximum $\\Pi_{i=1}^t |{\\cal\n  R}_i|$ over all ${\\cal R}=\\{{\\cal R}_1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.01735","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"}