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,ℓ 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,ℓ) be the restriction to subtrees of depth at most ℓ. It was proved by Landau that f_T(1,2)=n and proved by Sands et al. that f_T(2)=n where |V(T)|=n. Here we consider f_T(k) and f_T(k,ℓ) 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,ℓ), i.e., when T is a random tournament.