such that for all $I\in \mathfrak{P}$, $$\biggl|\frac{ \tilde{\tau}_s X_I}{(2 \tilde{\delta} \tilde{\mu}_s)^{1/2}} - 1 \biggr| < \tilde{\varepsilon}\, ,$$ and (b) for all $J\in \mathfrak{P}^c$, $$\frac{\tilde{\tau}_s X_J}{(2 \tilde{\delta} \tilde{\mu}_s)^{1/2}} \leq \tilde{\varepsilon}\, .$$ Then the conditional probability of $F$ given $|E_s| \geq (1+\tilde{\delta}) \tilde{\mu}_s$ tends to $1$ as $n\to \infty$.

Essentially, Theorem \[thm2\] produces a maximal clique set, and whose entries sum up to $(2 \tilde{\delta} \tilde{\mu}_s)^{1/2}$, up to lower order corrections. This creates all the ‘extra edges’ in the graph. As in Theorem \[thm1\], this set will contain almost all the edges not expected to be the in unconditional $s$-graded model, and the rest of graph will look unaltered, at least in the scale of $\sqrt{\tilde{\mu}_s}$. This theorem will also be instrumental in proving \[thmLDP\]; that proof will appear in Section 7.

Outline of the Proof
====================

Before embarking on a proper proof, we sketch the main ideas required. The first step is to show that the $s$-graded model is, in fact, a good approximation for the random geometric graph. To do so, we first show that we can approximate any convex subset $S$ of a ball of diameter $r$ from both the inside and the outside by a union of $A_I$’s. 