V$, define $$\psi(v) = {\mathbb{P}}(v \in {\bm{S}}), \quad 
   \psi_{\partial}(v) = {\mathbb{P}}(v \in \partial {\bm{S}}).$$ Our first parameter $\partial^+$ is $$\label{partial-plus}
 \partial^+ = \partial^+(\mathscr{S}) =
  \max_{S\in\mathscr{S}} |\partial S|^{\min\{ |S|,\, |\partial S|\}}.$$ Let $\psi_{\min} := \min_{v \in V} \psi(v)$ and define our second parameter $\Psi$ by $$\label{Psi}
 \Psi = \Psi(\mathscr{S}, \psi) = \max_{v \in V} \frac{\psi_\partial(v)}
                                         {\psi(v)}.$$ These first two parameters are in some sense less important than the third parameter since they are essentially used as crude estimates for quantities that we do not aim to control too precisely.

For the third parameter we require some terminology. Given $A \subseteq V$ and $X\in \Omega$, write $X|_A$ for the configuration $X$ restricted to $A$. Consider a configuration $c\in [q]^S$. A colour used by $c$ is called *free* with respect to $X,S$ if it does not appear in $X|_{\partial S}$. Write $f(X,S,c)$ for the number of free colours in $c$ with respect to $X,S$. For our third parameter, we first define for each positive integer $f$ $$\mu^+_{X,S,f}  =
   \max \left\{
    \frac{\mu_{X,S}(c)}{|S| - f} : c\in [q]^S,\,\, f(X,S,c)=f\right\},$$ where the maximum over an empty set is defined to be zero. We set $$\label{mu-plus}
 \mu^+ = \mu^+(\mathscr{S}) = \max_{S\in\mathscr{S}}\, \max_{X\in \Omega} \,  
    \max_{f=0,\ldots, |S|-1}\, 
 \mu^+_{X,S,f}.$$ 