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- Publisher
- Springer Journals
- Copyright
- Copyright © 2016 by The Author(s)
- Subject
- Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
- ISSN
- 0025-5858
- eISSN
- 1865-8784
- DOI
- 10.1007/s12188-016-0160-3
- Publisher site
- See Article on Publisher Site
Abstract
Stein (Pac J Math 59:567–575, 1975) proposed the following conjecture: if the edge set of $$K_{n,n}$$ K n , n is partitioned into n sets, each of size n, then there is a partial rainbow matching of size $$n-1$$ n - 1 . He proved that there is a partial rainbow matching of size $$n(1-\frac{D_n}{n!})$$ n ( 1 - D n n ! ) , where $$D_n$$ D n is the number of derangements of [n]. This means that there is a partial rainbow matching of size about $$(1- \frac{1}{e})n$$ ( 1 - 1 e ) n . Using a topological version of Hall’s theorem we improve this bound to $$\frac{2}{3}n$$ 2 3 n .
Journal
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals
Published: Jan 4, 2017