The singleton Bird family is generalized nice

Given: The singleton family $\{\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{d}\}$.

The singleton family $\{\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{d}\}$ has property $(\ast )$, with its special-vertex comb trigger in co-Bird-free graphs (The singleton Bird family has property (*)).

The singleton family $\{\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{d}\}$ is leaf-reducible: deleting the added leaf $w$ from Bird gives the bull, and the reduced singleton family has the Erdős-Hajnal property (The $E$-graph and Bird singleton families are leaf-reducible).

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

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, and leaf-reducibility asks that deleting one leaf from one member produce a family with the Erdős-Hajnal property (Property (*) for a finite graph family, Leaf-reducible finite graph families).

co-Bird is the complement of the Bird graph (The Bird graph and co-Bird).