Path-monochromatic bounded depth rooted trees in (random) tournaments

An edge-colored rooted directed tree (aka arborescence) is path-monochromatic if every path in it is monochromatic. Let $k,\mathrm{\ell }$ be positive integers. For a tournament $T$, let $f_T(k)$ be the largest integer such that every $k$-edge coloring of $T$ has a path-monochromatic subtree with at least $f_T(k)$ vertices and let $f_T(k,\mathrm{\ell })$ be the restriction to subtrees of depth at most $\mathrm{\ell }$. It was proved by Landau that $f_T(1,2)=n$ and proved by Sands et al. that $f_T(2)=n$ where $\mathrm{\mid }V(T)\mathrm{\mid }=n$. Here we consider $f_T(k)$ and $f_T(k,\mathrm{\ell })$ in more generality, determine their extremal values in most cases, and in fact in all cases assuming the Caccetta-Häggkvist Conjecture. We also study the typical value of $f_T(k)$ and $f_T(k,\mathrm{\ell })$, i.e., when $T$ is a random tournament.