= \mathbb{E}(|E_s|)$ and $\tau = \nu (r/2)^d$. Let $\tilde{\tau}_s$ be the cardinality of a maximal clique set (as defined in ). Then $E\subseteq E_s$, and there exist constants $C$ and $s_0$, depending only on the dimension and the chosen norm of the torus, such that if $s\ge s_0$ then $$\mu \leq \tilde{\mu}_s \leq \mu \left(1 + \frac{C}{s}\right)$$ and $$m^d \tau \leq \tilde{\tau}_s \leq m^d \tau \left(1 + \frac{C}{s}\right)$$ Furthermore, both $|N_I|$ and $\tilde{\tau}_s$ are uniformly bounded in $n$.

Pick an arbitrary $I$ and consider $\mathfrak{U}(N_I)$. By definition of $d$ and $s$, this set includes a ball of radius $r$ around $\textit{any}$ point in $A_I$. Therefore, any pair $(v,w) \in E$ must also be in $E_s$, giving the first stipulation. Since this inclusion holds for any configuration of the underlying Poisson Point process, this also gives $\mu \leq \tilde{\mu}_s$.

Now, let $\rho$ be the diameter of the unit cube under the norm $\| \cdot \|$. Then, for any $x \in A_I$ and $y \in \mathfrak{U}(N_I)$, $$\| x - y\| \leq \frac{rs}{s-r} + \frac{2 \rho}{m}$$ Therefore, $\mathfrak{U}(N_I)$ is contained in a ball of radius $r + 3\rho/m$ around any point in $A_I$, which implies that $$|N_I| = m^d \lambda\left(\mathfrak{U}(N_I)\right) \leq \nu m^d 