  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.
