A Further Extension of Rödl's Theorem

Fix $\varepsilon>0$ and a nonnull graph $H$. A well-known theorem of Rödl from the 80s says that every $G$ with no induced copy $H$ contains linear-sized $\varepsilon$-restricted set $S\subseteq V(G)$, which means $S$ induces subgraph maximum degree at most $\varepsilon |S|$ in or its complement. There are two extensions this result:
 
 quantitatively, Nikiforov (and later Fox Sudakov) relaxed condition "no $H$" into "at $\kappa|G|^{|H|}$ copies for some $\kappa>0$" depending on $\varepsilon$; and
 qualitatively, Chudnovsky, Scott, Seymour, Spirkl recently showed there exists $N>0$ $\varepsilon$ such is $(N,\varepsilon)$-restricted, $V(G)$ has partition $N$ subsets $\varepsilon$-restricted.
 natural common generalization these asserts $(N,\varepsilon)$-restricted $\kappa,N>0$. This unfortunately false, but we prove $\varepsilon>0$, $\kappa$ still exist so $d\ge0$, $\kappa d^{\vert H\vert}$ an least $\vert G\vert-d$ vertices. unifies aforementioned theorems, optimal up to$\kappa$ value $d$.