I am looking about the following Ramseyish theorem with additional structure on tournament (see ).
For all number of colors $k$, there exists $n=T(k)$ such that all tournaments of size $n$ colored with $k$ colors, there exists necessary a path of length $2$ monochromatic.
It is a direct consequence of Ramsey theorem when applied to triangle on the underlying colored graph. Now, I am looking for estimate on $T(k)$, that is the minimum size such that we are certain that a path of length $2$ exists. Remarks that $T(k)$ has to be at least larger than $k$. Indeed, otherwise you can associate to each vertices a unique color that you use to label its outgoing edges.
Actually, I don't really need the precise value but simply to know if it is polynomial or exponential. Note that the same question for coloring numbers in $[n]$ (i.e. the tournament is a full order) sounds already non trivial, or maybe I miss something?
: Wikipedia page for tournament https://en.wikipedia.org/wiki/Tournament_(graph_theory)#Ramsey_theory