What is the smallest number $n$ such that every two-colouring of the edges of $K_n$ contains a (not necessarily induced) path on 3 vertices?
If it is a path on 3 vertices , $K_3$ works as it is garanted that two of the three edges have the same color, well then they form a path on three vertices. If what we want is a path of length $3$, my guess is $5$. Look at a vertex $u \in V(K_n)$, there are 4 edges connect to it. It is guarantee that at least two of the edges $e_1,e_2$ connected to it have the same color. So the edges connected to $e_1$ and $e_2$ not via $u$ must be color the other color. But then that gives a path of length 3 in the other color.