A wonderful anticonnected complete-or-sparse blockade yields a restricted subgraph or a large anticomplete pair

Apply [L1] to the blockade $\mathcal{B}$. If it yields a $y^4$-restricted induced subgraph on at least $w$ vertices, then outcome 1 of the present lemma holds immediately.

We may therefore assume that [L1] yields an index $i\in [\mathrm{\ell }]$ for which at most $y\mathrm{\mid }G\mathrm{\mid }$ vertices outside $V(\mathcal{B})$ are mixed on $B_i$. Let $M$ be that exceptional set of mixed outside vertices. Every outside vertex with a neighbour in $B_i$ but not in $M$ has at least $\mathrm{\mid }{B}_i\mathrm{\mid }\mathrm{/}2$ neighbours in $B_i$. Since every vertex of $B_i$ has total degree at most $y\mathrm{\mid }G\mathrm{\mid }$ by [L2], the number of outside vertices with at least $\mathrm{\mid }{B}_i\mathrm{\mid }\mathrm{/}2$ neighbours in $B_i$ is at most $2y\mathrm{\mid }G\mathrm{\mid }$.

Let $Y$ be the set of vertices in $V(G)\setminus V(\mathcal{B})$ that have no neighbours in $B_i$, and let $X:=B_i$. By step 2.1, $\mathrm{\mid }Y\mathrm{\mid }\ge \mathrm{\mid }G\mathrm{\mid }-\mathrm{\mid }V(\mathcal{B})\mathrm{\mid }-\mathrm{\mid }M\mathrm{\mid }-2y\mathrm{\mid }G\mathrm{\mid }\ge \mathrm{\mid }G\mathrm{\mid }-y\mathrm{\mid }G\mathrm{\mid }-y\mathrm{\mid }G\mathrm{\mid }-2y\mathrm{\mid }G\mathrm{\mid }=(1-4y)\mathrm{\mid }G\mathrm{\mid }$. By construction there are no edges between $X$ and $Y$, so [L3] gives that $Y$ is anticomplete to $X$. Because all blocks have size $w$, we also have $\mathrm{\mid }X\mathrm{\mid }=\mathrm{\mid }{B}_i\mathrm{\mid }=w$. Thus outcome 2 holds.

Steps 1.1 and 3.1 prove that one of the two stated outcomes must occur.