1 - \frac{\xi}{\log n}\right\} \cap  \left\{\frac{1}{q} \displaystyle \sum_{I \in \mathfrak{I}} Y_I  \leq \log(q/w) -1 +
\xi\right\}\, ,$$ where $\xi$ is an arbitrary positive constant. By comparing a lower bound on the probability of $L$ to an upper bound on the probability of the [*complement*]{} of the above event, we can show that this event occurs with high probability in the $s$-graded model conditional on $L$.

We now have a set of indices $\mathfrak{I}$ which satisfies both a quadratic lower bound and a convex upper bound with high probability. From here on, the analysis is completely deterministic, characterizing configurations that satisfy the simultaneous inequalities above. We wish to look at the largest elements of $\mathfrak{I}$. Specifically, we wish to take the smallest set that includes roughly $q$ vertices of the Poisson point process. To make this precise, define $$V(A) := \frac{1}{q} \displaystyle \sum_{I \in A} X_I \, ,$$ for any subset of indices $A$. Now, order the elements of $\mathfrak{I}$ by size, so that $X_1$ is the largest, $X_2$ is the second largest, etc. Letting $\mathfrak{T}_k = \{X_1, X_2, \dots X_k\}$, define $$\mathfrak{T} := \text{ The first } \mathfrak{T}_k \text{ such that }  V(\mathfrak{T}_k) > 1 - \frac{2 \xi}{\log n}\, .$$ 