Generalized nice finite graph families

Let $\mathcal{F}$ be a finite family of finite graphs, and write

$\bar{\mathcal{F}}:=\{\bar{H}:H\in \mathcal{F}\}$

for the family of complements (Graph isomorphisms, automorphisms and graph complements).

We say that $\mathcal{F}$ is generalized nice if there exist real constants

$c_1\ge 3,c_2\ge 8,c_3,c_4,c_5,c_8>0,c_6\ge 1,c_7\ge 4$

such that for every $\bar{\mathcal{F}}$-free graph $G$ ($H$-free and $\mathcal{F}$-free graphs under the induced-subgraph convention) and every $ϵ\in (0,\frac{1}{2})$, at least one of the following holds:

1. $G$ has an $(\lceil {ϵ}^{-1}\rceil ,{ϵ}^{c_1}\mathrm{\mid }G\mathrm{\mid })$-blockade (Blockades, their length, their width, and their support) whose distinct block pairs are either complete or weakly ${ϵ}^{c_2}$-sparse (Sparsity of one vertex set to another, and weak sparsity of a pair);
2. $G$ has a clique or stable set of size at least $({ϵ}^{c_3}\mathrm{\mid }G\mathrm{\mid }{)}^{c_4}$ (Cliques, stable sets, the clique number $\omega (G)$ and stability number $\alpha (G)$);
3. $G$ has a complete or anticomplete $(k,\mathrm{\mid }G\mathrm{\mid }\mathrm{/}{k}^{c_5})$-blockade with $k\ge {ϵ}^{-c_6}$; or
4. $G$ has an ${ϵ}^{c_7}$-restricted induced subgraph of size at least ${ϵ}^{c_8}\mathrm{\mid }G\mathrm{\mid }$ ($c$-sparse, $c$-dense and $c$-restricted vertex sets).

This is the Section 3 replacement for the earlier "nice" condition: the first alternative still produces a long blockade, but the other three alternatives already package the three reduction outcomes that will be iterated later on the page.