then we are done by induction, so we may assume that $G$ is [*reduced*]{}. Here, the reduction operations depend on $J$ (for instance, removing vertices of degree-$1$ and suppressing vertices of degree-$2$ if $J$ is $C_3$). Finally, we partition the vertex set $V(G)$ of our reduced graph $G$ into sets that induce connected subgraphs of diameter proportional to $\log k$; in fact these subgraphs can be chosen to approximately equal balls of some uniform radius $r=O(\log k)$. For simplicity, we call them *balls* in the remainder of this proof sketch. Using that small $J$-expansions do not occur as a subgraph, we then need to derive that each ball grows exponentially, in the sense that the number of vertices on its boundary is exponential in $r$. This ensures that the boundary of each ball has size $\Omega(k^{1+\epsilon})$, for some $\epsilon$ depending on the size of $J$ but independent of $k$. We also need that sufficiently many vertices on the boundary have a neighbour in some other ball, and that between any two balls there are only ‘few’ edges. This will ensure that every ball sends an edge to $\Omega(k^{1+\epsilon})$ distinct other balls. Once this has been established, we can contract each 