Definition ($P_A$). Let $E\subset {\mathbb{Z}}^n$ and let $A\subset [n]$. Then we write $P_{A}E$ for the set of all $u\in {\mathbb{Z}}^A$ such that there exists $v\in {\mathbb{Z}}^{[n]\setminus A}$ such that $[u,v]\in E$, where $[u,v]$ is $u$ suitably intertwined with $v$ (i.e. $u\cup v$ as functions).