Complete or anticomplete blockade hypotheses force an $ϵ$-restricted induced subgraph

Let $q$ be maximal subject to the existence of a pure blockade $(A_1,\dots ,A_q)$ whose pattern graph is $P_4$-free, every block has size at least ${ϵ}^{3a}\mathrm{\mid }G\mathrm{\mid }$, and ${\sum }_{i=1}^q\mathrm{\mid }{A}_i{\mathrm{\mid }}^{1\mathrm{/}a}\ge \mathrm{\mid }G{\mathrm{\mid }}^{1\mathrm{/}a}$. By A maximal pure blockade with large total $a$-mass must already have at least ${ϵ}^{-2}$ blocks, one has $q\ge {ϵ}^{-2}$.

By Pure blockades with $P_4$-free patterns contain complete or anticomplete subblockades of square-root length, this blockade has a complete or anticomplete subblockade indexed by a set $I\subseteq [q]$ with $\mathrm{\mid }I\mathrm{\mid }\ge \sqrt{q}\ge {ϵ}^{-1}$. For each $i\in I$, choose $S_i\subseteq A_i$ with $\mathrm{\mid }{S}_i\mathrm{\mid }=\lceil {ϵ}^{3a}\mathrm{\mid }G\mathrm{\mid }\rceil$, and put $S:={\bigcup }_{i\in I}{S}_i$. Then $\mathrm{\mid }S\mathrm{\mid }=\mathrm{\mid }I\mathrm{\mid }\lceil {ϵ}^{3a}\mathrm{\mid }G\mathrm{\mid }\rceil \ge {ϵ}^{3a}\mathrm{\mid }G\mathrm{\mid }$.

If the chosen subblockade is anticomplete, then every vertex of $S_i$ has neighbors in $S$ only inside $S_i$, so its degree in $G[S]$ is at most $\mathrm{\mid }{S}_i\mathrm{\mid }-1\le \mathrm{\mid }S\mathrm{\mid }\mathrm{/}\mathrm{\mid }I\mathrm{\mid }\le ϵ\mathrm{\mid }S\mathrm{\mid }$. Hence $S$ is $ϵ$-sparse, and therefore $ϵ$-restricted.

If the chosen subblockade is complete, then in the complement $\bar{G}[S]$ every vertex of $S_i$ has neighbors only inside $S_i$, so the same estimate shows that $\bar{G}[S]$ is $ϵ$-sparse. Therefore $G[S]$ is $ϵ$-dense, and again $ϵ$-restricted.

In either case $G$ has an $ϵ$-restricted induced subgraph on at least ${ϵ}^{3a}\mathrm{\mid }G\mathrm{\mid }$ vertices, namely $G[S]$.