Long $A$-$B$-paths have the edge-Erd\H os-Pósa property

Abstract: For a fixed integer $\ell$ a path is long if its length is at least $\ell$. We prove that for all integers $k$ and $\ell$ there is a number $f(k,\ell)$ such that for every graph $G$ and vertex sets $A,B$ the graph $G$ either contains $k$ edge-disjoint long $A$-$B$-paths or it contains an edge set $F$ of size $|F|\leq f(k,\ell)$ that meets every long $A$-$B$-path. This is the edge analogue of a theorem of Montejano and Neumann-Lara (1984). We also prove a similar result for long $A$-paths and long $\mathcal{S}$-paths.