A digraph is $k$-linked if for any two disjoint sets of vertices $\{x_1,\dots ,x_k\}$ and $\{y_1,\dots ,y_k\}$ there are vertex disjoint paths $P_1,\dots ,P_k$ such that $P_i$ is directed from $x_i$ to $y_i$ for $i=1,\dots ,k$. Pokrovskiy in 2015 proved that every strongly $452k$-connected tournament is $k$-linked. In this paper, we significantly reduce this connectivity bound and show that any $(40k-31)$-connected tournament is $k$-linked.

中文翻译：

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一种改进的线性连接性，使锦标赛高度链接

一个有向图是 $克$-linked if 对于任何两个不相交的顶点集 $\{X_1,\dots ,X_{克}\}$ 和 $\{{是}_1,\dots ,{是}_{克}\}$ 存在顶点不相交的路径 ${磷}_1,\dots ,{磷}_{克}$ 以至于 ${磷}_{\mathrm{一世}}$ 来自 $X_{\mathrm{一世}}$ 至 ${是}_{\mathrm{一世}}$ 为了 $\mathrm{一世}=1,\dots ,克$. Pokrovskiy 在 2015 年证明，每个强烈$452克$-连接锦标赛是 $克$-链接。在本文中，我们显着降低了这种连通性界限，并表明任何$(40克-31)$-连接锦标赛是 $克$-链接。