The Bird graph has the Erdős-Hajnal property

Given: The singleton family $\{\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{d}\}$ and its class of Bird-free finite graphs.

The singleton family $\{\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{d}\}$ is generalized nice (The singleton Bird family is generalized nice).

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

The singleton family $\{\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{d}\}$ is wonderful (The $E$-graph and the Bird graph are wonderful).

Every generalized nice, leaf-reducible, wonderful finite family has the Erdős-Hajnal property (Leaf-reducible wonderful generalized nice finite families have the Erdős-Hajnal property).

A positive real $ϵ$ is an Erdős-Hajnal constant for a hereditary class $\mathcal{C}$ when every nonempty $G\in \mathcal{C}$ satisfies $\mathrm{hom}(G)\ge \mathrm{\mid }V(G){\mathrm{\mid }}^{ϵ}$; a graph $H$ has the Erdős-Hajnal property when its class of $H$-free graphs has such a constant, and $\mathrm{hom}(G)=\max \{\omega (G),\alpha (G)\}$ is the size of the largest clique or stable set of $G$ (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class, Homogeneous vertex sets and the homogeneous number $\mathrm{hom}(G)=\max \{\omega (G),\alpha (G)\}$).

Graph $G$ is $H$-free when it has no induced copy of $H$ ($H$-free and $\mathcal{F}$-free graphs under the induced-subgraph convention), and the class of $H$-free graphs is hereditary for every finite graph $H$ (Every class defined by forbidden induced subgraphs is hereditary).

The Bird graph has vertex set $\{x_1,x_2,x_3,y,z,w\}$ and edge set $\{x_1{x}_2,x_2{x}_3,x_1{x}_3,x_{1}y,x_{2}z,yw\}$ (The Bird graph and co-Bird).