Nikiforov: for every $H$ and every $ϵ\in (0,\frac{1}{2})$ there is $\delta >0$ such that every graph $G$ with ${\mathrm{ind}}_H(G)<(\delta \mathrm{\mid }V(G)\mathrm{\mid }{)}^{\mathrm{\mid }V(H)\mathrm{\mid }}$ has an $ϵ$-restricted vertex set of size at least $\delta \mathrm{\mid }V(G)\mathrm{\mid }$

Set $\eta :=ϵ\mathrm{/}4$ and choose $\gamma >0$ so small that $\gamma \le ϵ\mathrm{/}8$ and $\gamma \le \gamma (H,\eta )$ from [L2]. Then $(\eta +\gamma )\mathrm{/}(1-\gamma )\le ϵ$, and the dense trimming parameter from [L3] is also at most $ϵ$, because $\eta =ϵ\mathrm{/}4$, $\gamma \le ϵ\mathrm{/}8$, and $ϵ<1\mathrm{/}2$.

If $h=0$, the induced-copy hypothesis is never satisfied: both sides of its displayed inequality are $1$. Thus suppose $h\ge 1$. Let ${\delta }_1:={\delta }_1(\gamma )$ be the constant of [L1], let $c:=c(H,\eta )$ and $N:=N(H,\eta )$ be the constants of [L2], put $N^{\mathrm{\prime }}:=\max \{N,1\}$, and set $\delta :=\min \{{\delta }_1{c}^{1\mathrm{/}h},(1-\gamma ){\delta }_1,{\delta }_1\mathrm{/}{N}^{\mathrm{\prime }},1\}\mathrm{.}$ Then $\delta >0$ and depends only on $H$ and $ϵ$.

Now let $G$ be a nonempty graph on $n$ vertices with ${\mathrm{ind}}_H(G)<(\delta n{)}^h$. If $n<N^{\mathrm{\prime }}\mathrm{/}{\delta }_1$, then any singleton $X$ is $0$-restricted by [L4] and satisfies $\mathrm{\mid }X\mathrm{\mid }=1>\delta n$, because $\delta \le {\delta }_1\mathrm{/}{N}^{\mathrm{\prime }}$. Hence suppose $n\ge N^{\mathrm{\prime }}\mathrm{/}{\delta }_1$. By [L1] choose $W\subseteq V(G)$ with $\mathrm{\mid }W\mathrm{\mid }\ge {\delta }_{1}n\ge N^{\mathrm{\prime }}\ge N$ such that $(W,W)$ is $\gamma$-regular.

If $\eta \le d_G(W,W)\le 1-\eta$, then [L2] gives ${\mathrm{ind}}_H(G)\ge c\mathrm{\mid }W{\mathrm{\mid }}^h\ge c{\delta }_1^h{n}^h\ge (\delta n{)}^h$, contrary to the hypothesis on $G$. Therefore either $d_G(W,W)<\eta$ or $d_G(W,W)>1-\eta$.

In the first case, the low-density trimming lemma in [L3] yields a subset $W^{\mathrm{\prime }}\subseteq W$ with $\mathrm{\mid }{W}^{\mathrm{\prime }}\mathrm{\mid }>(1-\gamma )\mathrm{\mid }W\mathrm{\mid }\ge \delta n$ that is $ϵ$-sparse by step 1.1. In the second case, the high-density trimming lemma in [L3] yields a subset $W^{\mathrm{\prime }}\subseteq W$ with the same size bound that is $ϵ$-dense. In either case $W^{\mathrm{\prime }}$ is $ϵ$-restricted.

Step 3.1 handles small $n$, and steps 4.1 and 5.1 handle all remaining cases, so every graph satisfying the induced-copy bound has an $ϵ$-restricted set of size at least $\delta \mathrm{\mid }V(G)\mathrm{\mid }$.