Wonderful 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 its family of complements (Graph isomorphisms, automorphisms and graph complements).

We say that $\mathcal{F}$ is wonderful if there exists a real constant $a\ge 6$ such that the following holds for every $y\in (0,\frac{1}{2})$ and every $\bar{\mathcal{F}}$-free graph $G$ ($H$-free and $\mathcal{F}$-free graphs under the induced-subgraph convention).

Suppose that $\mathcal{B}=(B_1,\dots ,B_{\mathrm{\ell }})$ is an $(\mathrm{\ell },w)$-blockade in $G$ (Blockades, their length, their width, and their support) with $\mathrm{\ell }\ge y^{-a}$, that all blocks have the same size, that every block $B_i$ is anticonnected (Anticonnected graphs and anticonnected components), and that for every distinct $i,j\in [\mathrm{\ell }]$ either

• $B_i$ is complete to $B_j$ (Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs), or
• $B_i$ is $y^a$-sparse to $B_j$ and $B_j$ is $y^a$-sparse to $B_i$ (Sparsity of one vertex set to another, and weak sparsity of a pair).

Then at least one of the following conclusions holds:

1. $G$ has a $y^4$-restricted induced subgraph of size at least $w$ ($c$-sparse, $c$-dense and $c$-restricted vertex sets).
2. There exists $i\in [\mathrm{\ell }]$ such that at most $y\mathrm{\mid }V(G)\mathrm{\mid }$ vertices $v\in V(G)\setminus V(\mathcal{B})$ satisfy $0<\mathrm{\mid }{N}_G(v)\cap B_i\mathrm{\mid }<\frac{1}{2}\mathrm{\mid }{B}_i\mathrm{\mid }\mathrm{.}$

This item fixes the symmetric reading of the source phrase "complete or $y^a$-sparse" that the later proof actually uses: when a pair of blocks is not complete, each block is sparse to the other.