The singleton $E$-graph family is generalized nice

Given: The singleton family $\{E\}$ and the complement family $\overline{\{E\}}=\{\mathrm{c}\mathrm{o}\text{-}E\}$ of the published generalized-niceness convention.

The singleton family $\{E\}$ has property $(\ast )$ (The singleton family $\{E\}$ has property (*)).

The family $\{H_5,\mathrm{c}\mathrm{o}\text{-}E\}$ has the Erdős-Hajnal property, so it has an Erdős-Hajnal constant (The family consisting of $H_5$ and co-$E$ has the Erdős–Hajnal property).

Leaf/co-leaf transfer: if $\mathcal{F}$ is a finite family, $H_1\in \mathcal{F}$ has a leaf $v$, $H_2\in \mathcal{F}$ has a co-leaf $w$, and the two modified families are $\{H_1\setminus \{v\}\}\cup (\mathcal{F}\setminus \{H_1\})$ and $\{H_2\setminus \{w\}\}\cup (\mathcal{F}\setminus \{H_2\})$, then the Erdős-Hajnal property of both modified families implies it for $\mathcal{F}$ (Deleting a leaf and a co-leaf preserves the Erdős-Hajnal property of a finite forbidden family).

In every co-$E$-free graph, every special-vertex comb of the property-$(\ast )$ trigger admits the $\{H_5,\mathrm{c}\mathrm{o}\text{-}E\}$ structural partition: each block splits as $B_i=X_i\cup Y_i$ with $Y_i$ $\{H_5,\mathrm{c}\mathrm{o}\text{-}E\}$-free and $X_i$ carrying a nonempty-block pure blockade partition whose pattern is $\{H_5,\mathrm{c}\mathrm{o}\text{-}E\}$-free and whose blocks are pure to every vertex of the other comb blocks (A special-vertex comb in a co-$E$-free graph admits the $\{H_5,\mathrm{c}\mathrm{o}\text{-}E\}$ structural partition).

Special-vertex-local criterion: if finite families ${\mathcal{F}}_1,{\mathcal{F}}_2$ have a common Erdős-Hajnal constant $c\in (0,1]$ and every special-vertex comb in every $\bar{\mathcal{H}}$-free graph admits a partition with clauses (1) and (2.1)--(2.3) of the structural comb partition, then $\mathcal{H}$ has property $(\ast )$ (The special-vertex-local structural-partition criterion implies property (*)).

If a finite family has property $(\ast )$ and is leaf-reducible, then it is generalized nice (Property (*) and leaf reducibility imply generalized niceness).

The singleton family $\{E\}$ is leaf-reducible (The $E$-graph and Bird singleton families are leaf-reducible).

Generalized niceness of a finite family $\mathcal{F}$ is the four-outcome schema quantified over $\bar{\mathcal{F}}$-free graphs (Generalized nice finite graph families).

Property $(\ast )$ for a finite family $\mathcal{F}$ is a condition on $\bar{\mathcal{F}}$-free graphs carrying the special-vertex comb trigger (Property (*) for a finite graph family).

The $E$-graph has edge set $\{p_1{p}_2,p_2{p}_3,p_3{p}_4,p_4{p}_5,p_{3}q\}$ on six vertices, and co-$E$ is its complement (The $E$-graph and co-$E$).

A vertex $v$ is a co-leaf of a graph $G$ when ${\mathrm{deg}}_G(v)=\mathrm{\mid }V(G)\mathrm{\mid }-2$, equivalently when $v$ is adjacent to every other vertex except one (Co-leaves of a finite graph).

If $ϵ$ is an Erdős-Hajnal constant for a hereditary class and $0<\delta \le ϵ$, then $\delta$ is one too (Every smaller positive exponent is again an Erdős–Hajnal constant).