{"paper":{"title":"On a conjecture of Stein","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dani Kotlar, Eli Berger, Ran Ziv, Ron Aharoni","submitted_at":"2016-05-06T15:57:07Z","abstract_excerpt":"Stein proposed the following conjecture: if the edge set of $K_{n,n}$ is partitioned into $n$ sets, each of size $n$, then there is a partial rainbow matching of size $n-1$. He proved that there is a partial rainbow matching of size $n(1-\\frac{D_n}{n!})$, where $D_n$ is the number of derangements of $[n]$. This means that there is a partial rainbow matching of size about $(1- \\frac{1}{e})n$. Using a topological version of Hall's theorem we improve this bound to $\\frac{2}{3}n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.01982","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"}