An iterative sparsification step for sparse $\overline{P_5}$-free graphs

Given: The constant $c=2^{-8}$, a parameter $x\in (0,c^5)$, and a $c^{16}$-sparse $\overline{P_5}$-free graph $G$ with $\mathrm{\mid }V(G)\mathrm{\mid }\ge x^{-7}$.

Lemma 5.4 of Nguyen, Scott, and Seymour's cited paper gives the displayed two outcomes with these constants and exponents. Its statement prints $\mathrm{\mid }G\mathrm{\mid }\ge y^{-7}$ before $y$ is bound; the proof shows that the intended hypothesis is $\mathrm{\mid }G\mathrm{\mid }\ge x^{-7}$ by using it to deduce $cx\mathrm{\mid }G\mathrm{\mid }\ge x^2\mathrm{\mid }G\mathrm{\mid }\ge x^{-5}$.

The source proof chooses a minimal threshold $y\in [cx,c^5]$, applies its preceding three-outcome sparse-blockade lemma, and rules out the deeper-sparsification branch by minimality. The remaining branches give the pure blockade in outcome 1 or the $x$-sparse blockade in outcome 2.