Generalized niceness yields four reduction outcomes

Fix constants from [L1], [L2], and [L4], and set $a_1:=a{c}_3$, $a_2:=c_4$, $a_4:=4$, $a_5:=h+4$, and $a_3:=\max \{a(c_1+5),a{c}_8,c_5,4d+1\}$. These choices depend only on $\mathcal{F}$.

If $\mathrm{\mid }G\mathrm{\mid }\le y^{-a_3}$, then any one-vertex induced subgraph of $G$ is $y^4$-restricted and has size at least $y^{a_3}\mathrm{\mid }G\mathrm{\mid }$. So outcome 2 holds.

Suppose $\bar{G}$ is $y$-sparse. Apply [L2] to the family $\mathcal{F}$ inside $\bar{G}$ with the parameter $b=4$. Either $\bar{G}$ has a $y^4$-restricted induced subgraph of size at least $y^{4d+1}\mathrm{\mid }G\mathrm{\mid }$, or there are disjoint sets $X,Y\subseteq V(G)$ with $\mathrm{\mid }X\mathrm{\mid }\ge y^{4d+1}\mathrm{\mid }G\mathrm{\mid }$, $\mathrm{\mid }Y\mathrm{\mid }\ge (1-hy)\mathrm{\mid }G\mathrm{\mid }$, and $Y$ anticomplete to $X$ in $\bar{G}$. By [L3], the restricted induced subgraph is also $y^4$-restricted in $G$, and the anticomplete pair in $\bar{G}$ is a complete pair in $G$. Since $a_3\ge 4d+1$ and $a_5\ge h$, this gives outcome 2 or outcome 4 in $G$.

We may therefore assume that $G$ itself is $y$-sparse. Put $ϵ:=y^a$, where $a$ is the witness from [L4]. Because $G$ is $\bar{\mathcal{F}}$-free, [L1] applies to $G$ and $ϵ$. If [L1] produces a clique or stable set of size $({ϵ}^{c_3}\mathrm{\mid }G\mathrm{\mid }{)}^{c_4}$, then this is exactly outcome 1 by the choice $a_1=a{c}_3$ and $a_2=c_4$. If [L1] produces a complete or anticomplete $(k,\mathrm{\mid }G\mathrm{\mid }\mathrm{/}{k}^{c_5})$-blockade with $k\ge {ϵ}^{-c_6}$, then $k\ge y^{-1}$ because $c_6\ge 1$, and $\mathrm{\mid }G\mathrm{\mid }\mathrm{/}{k}^{c_5}\ge \mathrm{\mid }G\mathrm{\mid }\mathrm{/}{k}^{a_3}$ because $a_3\ge c_5$, so outcome 3 holds. If [L1] produces an ${ϵ}^{c_7}$-restricted induced subgraph of size at least ${ϵ}^{c_8}\mathrm{\mid }G\mathrm{\mid }$, then ${ϵ}^{c_7}\le y^4$ and ${ϵ}^{c_8}=y^{a{c}_8}\ge y^{a_3}$, so outcome 2 holds. We are left only with the blockade alternative from [L1].

Thus $G$ has a blockade $\mathcal{A}=(A_1,\dots ,A_{\mathrm{\ell }})$ with $\mathrm{\ell }=\lceil {ϵ}^{-1}\rceil$, each $\mathrm{\mid }{A}_i\mathrm{\mid }\ge m:={ϵ}^{c_1}\mathrm{\mid }G\mathrm{\mid }$, and every distinct pair complete or weakly ${ϵ}^{c_2}$-sparse. Apply [L5] to obtain equal-sized subblocks $\mathcal{D}=(D_1,\dots ,D_{\mathrm{\ell }})$ with $\mathrm{\mid }{D}_i\mathrm{\mid }=q:=\lceil ϵ\lceil m\rceil \rceil$ and every noncomplete pair mutually ${ϵ}^{c_2-5}$-sparse. Then apply [L6] to $\mathcal{D}$. If [L6] yields a complete $(\mathrm{\ell },q\mathrm{/}{\mathrm{\ell }}^2)$-blockade, then $q\mathrm{/}{\mathrm{\ell }}^2\ge ϵm\cdot {ϵ}^4={ϵ}^{5}m=y^{a(c_1+5)}\mathrm{\mid }G\mathrm{\mid }\ge y^{a_3}\mathrm{\mid }G\mathrm{\mid }$, because $\mathrm{\ell }=\lceil {ϵ}^{-1}\rceil \le {ϵ}^{-2}$ for $ϵ\in (0,\frac{1}{2})$ and step 1.1 has $a_3\ge a(c_1+5)$. Since also $\mathrm{\ell }\ge {ϵ}^{-1}=y^{-a}\ge y^{-1}$, outcome 3 follows.

We may therefore assume [L6] yields anticonnected subsets $B_i\subseteq D_i$ of common size $w:=\lceil q\mathrm{/}\mathrm{\ell }\rceil$ such that every distinct pair is either complete or mutually $({ϵ}^{c_2-5}\mathrm{\ell })$-sparse. Because $\mathrm{\ell }\le {ϵ}^{-2}$, every noncomplete pair is in fact mutually ${ϵ}^{c_2-7}$-sparse. Also $w\ge q\mathrm{/}\mathrm{\ell }\ge {ϵ}^{3}m={ϵ}^{c_1+3}\mathrm{\mid }G\mathrm{\mid }=y^{a(c_1+3)}\mathrm{\mid }G\mathrm{\mid }\ge y^{a_3}\mathrm{\mid }G\mathrm{\mid }$, since $q\ge ϵm$ and $\mathrm{\ell }\le {ϵ}^{-2}$. Since step 2.1 fails, $\mathrm{\mid }G\mathrm{\mid }>y^{-a_3}$, and because $a_3\ge a(c_1+5)$ with $ϵ=y^a$, this gives $m={ϵ}^{c_1}\mathrm{\mid }G\mathrm{\mid }>y^{-5a}={ϵ}^{-5}$ and hence $q=\lceil ϵ\lceil m\rceil \rceil >ϵm>{ϵ}^{-4}\ge {ϵ}^{-2}\ge \mathrm{\ell }$. Therefore $\mathrm{\mid }V(\mathcal{B})\mathrm{\mid }={\sum }_{i=1}^{\mathrm{\ell }}\mathrm{\mid }{B}_i\mathrm{\mid }\le \mathrm{\ell }(q\mathrm{/}\mathrm{\ell }+1)=q+\mathrm{\ell }\le 2q\le 2\lceil m\rceil \le 4m\le {ϵ}^{c_1-2}\mathrm{\mid }G\mathrm{\mid }\le ϵ\mathrm{\mid }G\mathrm{\mid }\le y\mathrm{\mid }G\mathrm{\mid }$, where $\mathcal{B}:=(B_1,\dots ,B_{\mathrm{\ell }})$. Since $c_2\ge 8$, one has ${ϵ}^{c_2-7}=y^{a(c_2-7)}\le y^a$. Therefore the hypotheses of [L7] hold for $\mathcal{B}$.

Applying [L7] to $\mathcal{B}$ yields either a $y^4$-restricted induced subgraph of size at least $w$, giving outcome 2, or disjoint sets $X,Y\subseteq V(G)$ with $\mathrm{\mid }X\mathrm{\mid }=w\ge y^{a_3}\mathrm{\mid }G\mathrm{\mid }$, $\mathrm{\mid }Y\mathrm{\mid }\ge (1-4y)\mathrm{\mid }G\mathrm{\mid }\ge (1-a_{5}y)\mathrm{\mid }G\mathrm{\mid }$, and $Y$ anticomplete to $X$, giving outcome 4.

The cases in steps 2.1, 2.2, 2.3, and 5.1 exhaust all possibilities, so one of the four stated outcomes always holds.