Asymptotic structure. V. The coarse Menger conjecture in bounded path-width

Menger's theorem tells us that if $S,T$ are sets of vertices in a graph $G$, then (for $k\ge0$) either there are $k+1$ vertex-disjoint paths between $S$ and $T$, or there is a set of $k$ vertices separating $S$ and $T$. But what if we want the paths to be far apart, say at distance at least $c$? One might hope that we can find either $k+1$ paths pairwise far apart, or $k$ sets of bounded radius that separate $S$ and $T$, where the bound on the radius is some $\ell$ that depends only on $k,c$ (the ``coarse Menger conjecture''). We showed in an earlier paper that this is false for all $k\ge 2$ and $c\ge3$. To do so we gave a sequence of finite graphs, counterexamples for larger and larger values of $\ell$ with $k=2$, $c=3$. Our counterexamples contained subdivisions of uniform binary trees with arbitrarily large depth as subgraphs. Here we show that for any binary tree $T$, the coarse Menger conjecture is true for all graphs that contain no subdivision of $T$ as a subgraph, that is, it is true for graphs with bounded path-width (and, further, for graphs with bounded coarse path-width). This is perhaps surprising, since it is false for bounded tree-width.