Phase transitions in isoperimetric problems on the integers

Barber and Erde asked the following question: if $B$ generates ${\mathbb{Z}}^d$ as an additive group, then must the extremal sets for the vertex/edge-isoperimetric inequality on the Cayley graph $\mathrm{Cay}({\mathbb{Z}}^d,B)$ form a nested family? We answer this question negatively for both the vertex- and edge-isoperimetric inequalities, specifically in the case of $d=1$. The key is to show that the structure of the cylinder $\mathbb{Z}\times (\mathbb{Z}\mathrm{/}k\mathbb{Z})$ 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.