Next, we use the classical isodiametric inequality to show that the $A_I$’s associated with a maximal clique set approximate a ball of diameter $r$, in the sense of the Hausdorff metric. Armed with these facts, showing that Theorem \[thm2\] implies Theorem \[thm1\] is a matter of careful “epsilontics.”

We then turn to directly analyzing the $s$-graded model, conditioned on the event $$L := \{|E_s| \geq (1+\tilde{\delta}) \tilde{\mu}_s\}\, .$$ For notational convenience, let $q = \sqrt{2 \tilde{\delta} \tilde{\mu}_s}$, $w = \tilde{\tau}_s \mathcal{D}$. Define $$\mathfrak{I} = \{ I \in T : X_I > \max\{\mathcal{D}n^a, n^a\} \}\, ,$$ where $a$ is a small positive number. This is the set of “large” indices, in the sense that their associated $X_I$’s exceed their expected value by a fixed polynomial factor in $n$. Furthermore, define $$Y_I := X_I \left(\log (X_I /\mathcal{D}) -1\right) + \mathcal{D} \, ,$$ and $$Q(\mathfrak{I}) := \frac{2}{q^2}\biggl( \displaystyle \sum_{I \in \mathfrak{I}} {X_I \choose 2} + \frac{1}{2} \displaystyle \sum_{J \in N_I \cap
\mathfrak{I}} X_I X_J\biggr) \,.$$ The former quantity is an appropriately chosen convex function of the $X_I$’s, while the latter is a scaled version of the number of edges with both endpoints in the $A_I$’s associated with $\mathfrak{I}$. Consider the event $$\left\{Q(\mathfrak{I}) > 