Careful use of minimality and Jensen’s inequality can establish that $$V(\mathfrak{T}) \leq 1 + \phi(\mathfrak{T}) \, ,$$ where $\phi(\mathfrak{T})$ is bounded above by $\xi/3$. More importantly, moving from $\mathfrak{I}$ to $\mathfrak{T}$ does not force us to discard too many edges; formally, $$Q(\mathfrak{T}) \geq 1 - \psi(\mathfrak{T}) \, ,$$ with $\psi(\mathfrak{T})$ bounded above by $\xi$.

We now define $$\mathfrak{P} := \left\{ I \in \mathfrak{T} : X_I > \frac{\xi^{1/4} q}{\tilde{\tau}_s} \right\}\, .$$ If $\mathfrak{I}$ was the set of large indices, $\mathfrak{P}$ is the set of very large indices — that is, those whose associated $X_I$’s are commensurate with $q$. We can show that this set cannot have diameter that strictly exceeds $s$ without contradicting either the lower bound $Q(\mathfrak{T})$ or the upper bound on $V(\mathfrak{T})$. Furthermore, technical estimates ensure that $|\mathfrak{P}| \geq \tilde{\tau}_s$. As a set of diameter at most $s$ and cardinality at least $\tilde{\tau}_s$, $\mathfrak{P}$ must be a maximal clique set, by definition. Moreover, a quantitative version of Jensen’s inequality allows us to claim that all the $X_I$’s associated with this set are roughly equal. Thus, $\mathfrak{P}$ is the maximal clique set described in Theorem \[thm2\]. Finally, we can show that its complement is made up on $X_I$’s that vanish 