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On a conjecture of Stein
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg ( IF 0.4 ) Pub Date : 2017-01-04 , DOI: 10.1007/s12188-016-0160-3
Ron Aharoni , Eli Berger , Dani Kotlar , Ran Ziv

Stein (Pac J Math 59:567–575, 1975) proposed the following conjecture: if the edge set of $$K_{n,n}$$Kn,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-Dnn!), where $$D_n$$Dn is the number of derangements of [n]. This means that there is a partial rainbow matching of size about $$(1- \frac{1}{e})n$$(1-1e)n. Using a topological version of Hall’s theorem we improve this bound to $$\frac{2}{3}n$$23n.

中文翻译:

关于斯坦因的猜想

Stein (Pac J Math 59:567–575, 1975) 提出了以下猜想:如果 $$K_{n,n}$$Kn,n 的边集被划分为 n 个集,每个集的大小为 n,则有大小为 $$n-1$$$n-1 的部分彩虹匹配。他证明了存在大小为 $$n(1-\frac{D_n}{n!})$$n(1-Dnn!) 的部分彩虹匹配,其中 $$D_n$$Dn 是[n]。这意味着存在大小约为 $$(1- \frac{1}{e})n$$(1-1e)n 的部分彩虹匹配。使用霍尔定理的拓扑版本,我们将这个界限改进为 $$\frac{2}{3}n$$23n。
更新日期:2017-01-04
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