the geometry, we see that there are approximately $\sqrt{2 \delta \mu_n}$ vertices that are more or less equi-distributed inside $A$. Outside of this set, we would like to say that there are no other large cliques; unfortunately, \[thm1\] does not provide this result. Instead, we can only be sure that every other clique outside of the “exceptional" set $A$ has $o(\sqrt{\mu_n})$ vertices. We conjecture that the size of the largest clique disjoint from $A$ is of the same order as the size of the largest clique of an [*unconditional*]{} random geometric graph with the same values of $n$ and $r_n$. A combination of this conjecture and elementary estimates on Poisson Point Processes should be sufficient to establish a (suitably formalized) version of the claim ‘the remaining portion of the graph remains unchanged.’

The $s$-Graded Model
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Henceforth in the manuscript, we will suppress the subscript $n$ and write $\chi$, $\mu$, $\tau$ and $r$ instead of $\chi_n$, $\mu_n$, $\tau_n$ and $r_n$.

We now present an approximation of the random geometric model which allows us to replace the Poisson point process with a sequence of independent Poisson random variables. To do this, we first discretize space, and then produce a metric on 