Asymptotic structure. III. Excluding a fat tree

Robertson and Seymour proved that for every finite tree $H$, there exists $k$ such that every finite graph $G$ with no $H$ minor has path-width at most $k$; and conversely, for every integer $k$, there is a finite tree $H$ such that every finite graph $G$ with an $H$ minor has path-width more than $k$. If we (twice) replace ``path-width'' by ``line-width'', the same is true for infinite graphs $G$. We prove a ``coarse graph theory'' analogue, as follows. For every finite tree $H$ and every $c$, there exist $k,L,C$ such that every graph that does not contain $H$ as a $c$-fat minor admits an $(L,C)$-quasi-isonetry to a graph with line-width at most $k$; and conversely, for all $k,L,C$ there exist $c$ and a finite tree $H$ such that every graph that contains $H$ as a $c$-fat minor admits no $(L,C)$-quasi-isometry to a graph with line-width at most $k$.