$P_{k}$-freeness implies small dichromatic number

We propose a purely combinatorial quadratic time algorithm that for any $n$-vertex $P_{k}$-free tournament $T$, where $P_{k}$ is a directed path of length $k$, finds in $T$ a transitive subset of order $n^{\frac{c}{k\log(k)^{2}}}$. As a byproduct of our method, we obtain subcubic $O(n^{1-\frac{c}{k\log(k)^{2}}})$-approximation algorithm for the optimal acyclic coloring problem on $P_{k}$-free tournaments. Our results are tight up to the $\log(k)$-factor in the following sense: there exist infinite families of $P_{k}$-free tournaments with largest transitive subsets of order at most $n^{\frac{c\log(k)}{k}}$. As a corollary, we give tight asymptotic results regarding the so-called \textit{Erd\H{o}s-Hajnal coefficients} of directed paths. These are some of the first asymptotic results on these coefficients for infinite families of prime graphs.