Abstract
Stein (Pac J Math 59:567–575, 1975) 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 \(\frac{2}{3}n\).
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Aharoni, R., Berger, E., Kotlar, D. et al. On a conjecture of Stein. Abh. Math. Semin. Univ. Hambg. 87, 203–211 (2017). https://doi.org/10.1007/s12188-016-0160-3
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DOI: https://doi.org/10.1007/s12188-016-0160-3