Property (*) for a finite graph family

Let $\mathcal{F}$ be a finite family of finite graphs. We say that $\mathcal{F}$ has property $(\ast )$ if there exist constants $c_1,c_2,c_3>0$ such that the following holds for every $\bar{\mathcal{F}}$-free graph $G$, where $\bar{\mathcal{F}}:=\{\bar{H}:H\in \mathcal{F}\}$ is the family of graph complements (Graph isomorphisms, automorphisms and graph complements) ($H$-free and $\mathcal{F}$-free graphs under the induced-subgraph convention).

Suppose there is an $(\mathrm{\ell },w)$-comb $((a_i,B_i):i\in [\mathrm{\ell }])$ in $G$ (Combs in a graph) with $\mathrm{\ell },w\ge 4$, and suppose there is a vertex $v\in V(G)\setminus (\{a_i:i\in [\mathrm{\ell }]\}\cup {\bigcup }_{i=1}^{\mathrm{\ell }}{B}_i)$ such that $v$ is complete to ${\bigcup }_i{B}_i$ and anticomplete to $\{a_i:i\in [\mathrm{\ell }]\}$. Then at least one of the following holds:

1. $G$ has a clique or stable set of size at least $w^{c_1}$ (Cliques, stable sets, the clique number $\omega (G)$ and stability number $\alpha (G)$);
2. $G$ has a complete or anticomplete $(k,w\mathrm{/}{k}^{c_2})$-blockade for some real $k\ge {\mathrm{\ell }}^{c_3}$, where the real length threshold $k$ means that the blockade's integral length is at least $\lceil k\rceil$ (Blockades, their length, their width, and their support, Complete, anticomplete, pure, weakly sparse, and $x$-sparse blockades);
3. $G$ has a pure $(\mathrm{\ell },w\mathrm{/}{\mathrm{\ell }}^2)$-blockade.

This condition records exactly the three ways the special-vertex comb trigger can terminate the second sparsification round.