A Note on Powers of Paths in Tournaments

In this note we show that every tournament on $n$ vertices contains the $k$-th power of a directed path of length $n\mathrm{/}{2}^{6k+7}$, which improves upon the recent bound of Scott and Kor'{a}ndi of $n\mathrm{/}{2}^{2^{3k}}$. By doing so, we get an inverse exponential dependence on $k$, which is best possible as Yuster recently showed an upper bound of $kn\mathrm{/}{2}^{k\mathrm{/}2}$.