A coarse Gallai theorem

Abstract

We prove that there exist functions $f$ and $g$ such that for all positive integers $k$ and $d$, for every graph $G$ and every subset $A$ of the vertices of $G$, either $G$ contains $k$ $A$-paths such that vertices of different $A$-paths are at distance at least $d$ in $G$, or there exists a set $X$ of the vertices of $G$ with $|X|\le f(k)$ such that every $A$-path in $G$ contains a vertex of $B_G(X,g(k,d))$.