# Ramsey number for path in a colored tournament

I am looking about the following Ramseyish theorem with additional structure on tournament (see [1]).

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?

• Yes, it is coloring the edge and length is about number of edges as well.
– C.P.
Jul 7, 2018 at 8:51
• 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, 2018 at 9:04
• (Sorry, I posted this in my replies as en edit but it seems to have disappear ... the question is edited to take this remark into account)
– C.P.
Jul 7, 2018 at 9:04
• I didn't notice that you are talking about tournaments not complete graphs Jul 7, 2018 at 11:24