Let $\mathcal{G}_k$ denote the set of all graphs that contain two vertices $x,y$ and $k$ edge-disjoint $x-y$ paths.
Define $f(k)$ to be the maximum such that for every graph $G\in \mathcal{G}_k$ there are two vertices $x',y'$ with $f(k)$ independent $x'-y'$ paths in $G$.
Here, a set of paths is independent if none contains an internal vertex of another.
Are any lower bounds for $f(k)$ known in the literature?
In particular, I need $f(3)=3$ in an algorithm where I need to find some obstructions in a graph. This is not hard to prove, but I'm wondering whether it is known in the literature, maybe as a special case of a more general theorem.
As a different formulation, how large can you make $f(k)$ in the following statement?
If $G$ is an undirected graph containing two vertices $x,y$ and $k$ edge-disjoint $x-y$ paths, then $G$ contains two vertices $x',y'$ with $f(k)$ independent $x'-y'$ paths.