# Ramsey number for path in a colored tournament

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?

• No, it is not working, $T(k)$ has to be large. If $T(k)<k$ you have an obvious counter-example by associating each vertices to a unique color labeling its outgoing edges. – C.P. Jul 7 '18 at 9:04