  Barber and Erde asked the following question: if $B$ generates $\mathbb Z^n$
as an additive group, then must the extremal sets for the
vertex/edge-isoperimetric inequality on the Cayley graph
$\operatorname{Cay}(\mathbb Z^n,B)$ form a nested family? We answer this
question negatively for both the vertex- and edge-isoperimetric inequalities,
specifically in the case of $n=1$. The key is to show that the structure of the
cylinder $\mathbb Z\times(\mathbb Z/k\mathbb Z)$ can be mimicked in certain
Cayley graphs on $\mathbb Z$, leading to a phase transition. We do, however,
show that Barber--Erde's question for Cayley graphs on $\mathbb Z$ has a
positive answer if one is allowed to ignore finitely many sets.
