A new connectivity bound for a tournament to be highly linked

A digraph $D$ is $k$-linked if for any pair of two disjoint sets $\{x_{1},x_{2},\ldots,x_{k}\}$ and $\{y_{1},y_{2},\ldots,y_{k}\}$ of vertices in $D$, there exist vertex disjoint dipaths $P_{1},P_{2},\ldots,P_{k}$ such that $P_{i}$ is a dipath from $x_{i}$ to $y_{i}$ for each $i\in[k]$. Pokrovskiy (JCTB, 2015) confirmed a conjecture of K\"{u}hn et al. (Proc. Lond. Math. Soc., 2014) by verifying that every $452k$-connected tournament is $k$-linked. Meng et al. (Eur. J. Comb., 2021) improved this upper bound by showing that any $(40k-31)$-connected tournament is $k$-linked. In this paper, we show a better upper bound by proving that every $\lceil 12.5k-6\rceil$-connected tournament with minimum out-degree at least $21k-14$ is $k$-linked. Furthermore, we improve a key lemma that was first introduced by Pokrovskiy (JCTB, 2015) and later enhanced by Meng et al. (Eur. J. Comb., 2021).