The singleton Bird family has property (*)

Given: An arbitrary co-Bird-free finite graph $G$ and an arbitrary special-vertex comb in it, with special vertex $v$ complete to ${\bigcup }_i{B}_i$ and anticomplete to the teeth $\{a_i\}$.

The $E$-graph has the Erdős-Hajnal property: there is ${ϵ}_E>0$ such that every nonempty $E$-free graph has a clique or stable set of size at least $\mathrm{\mid }V(G){\mathrm{\mid }}^{{ϵ}_E}$ (The $E$-graph has the Erdős-Hajnal property).

Every positive exponent below an Erdős-Hajnal constant of a hereditary class is again one (Every smaller positive exponent is again an Erdős–Hajnal constant).

Let $((a_k,B_k):k\in [\mathrm{\ell }])$ be an $(\mathrm{\ell },w)$-comb in a finite simple co-Bird-free graph $G$, and let $v$ be outside all teeth and blocks, complete to every $B_k$ and anticomplete to every tooth. For every $i$ there are disjoint sets $X_i,Y_i$ with $B_i=X_i\cup Y_i$ such that $G[Y_i]$ is $E$-free, and $X_i$ has a partition into a nonempty ordered sequence $(A_1^i,\dots ,A_{t_i}^i)$ of nonempty sets that is a pure blockade, whose pattern is $E$-free, and such that each individual vertex of every other comb block is pure to each $A_j^i$ (A special-vertex co-Bird-free comb admits an E-free structural partition).

Special-vertex-local criterion: let ${\mathcal{F}}_1,{\mathcal{F}}_2$ have a common Erdős-Hajnal constant $c\in (0,1]$. Suppose that, in every $\bar{\mathcal{H}}$-free graph, every special-vertex comb occurring in the definition of property $(\ast )$ has a partition satisfying 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 (*)).

Property $(\ast )$ for a finite family $\mathcal{F}$ asks, for every $\bar{\mathcal{F}}$-free graph containing an $(\mathrm{\ell },w)$-comb with $\mathrm{\ell },w\ge 4$ and a vertex $v$ outside all teeth and blocks complete to ${\bigcup }_i{B}_i$ and anticomplete to $\{a_i\}$, that one of three listed outcomes hold with constants $c_1,c_2,c_3>0$ (Property (*) for a finite graph family).

The structural comb-partition clauses are: (1) $Y_i$ is ${\mathcal{F}}_1$-free; (2) $X_i$ has a nonempty-block pure-blockade partition whose pattern graph is ${\mathcal{F}}_2$-free; (3) every vertex of ${\bigcup }_{k\mathrm{\ne }i}{B}_k$ is pure to every block of that partition (The structural comb-partition hypothesis).

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

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

A graph is $H$-free when it has no induced copy of $H$, and $\mathcal{F}$-free when it is $H$-free for every $H\in \mathcal{F}$ ($H$-free and $\mathcal{F}$-free graphs under the induced-subgraph convention).

A graph $H$ has the Erdős-Hajnal property when its class of $H$-free graphs has an Erdős-Hajnal constant, and the same applies to a finite family through its family-free class (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class).