The structural comb-partition hypothesis

Let ${\mathcal{F}}_1,{\mathcal{F}}_2,\mathcal{H}$ be finite families of finite graphs, and suppose that ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have the Erdős–Hajnal property (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class). We say that $({\mathcal{F}}_1,{\mathcal{F}}_2;\mathcal{H})$ satisfies the structural comb-partition hypothesis if the following universal assertion holds.

For every $\bar{\mathcal{H}}$-free finite graph $G$ (Graph isomorphisms, automorphisms and graph complements, $H$-free and $\mathcal{F}$-free graphs under the induced-subgraph convention) and every $(\mathrm{\ell },w)$-comb $((a_i,B_i):i\in [\mathrm{\ell }])$ in $G$ with $\mathrm{\ell },w\ge 4$ (Combs in a graph), each $B_i$ has a partition such that:

1. $Y_i$ is ${\mathcal{F}}_1$-free;
2. $X_i$ has a partition $(A_1^i,\dots ,A_{t_i}^i)$ which is a pure blockade, its blocks being nonempty, whose pattern graph is ${\mathcal{F}}_2$-free (Complete, anticomplete, pure, weakly sparse, and $x$-sparse blockades, The pattern graph of a pure blockade); and
3. for every $j\in [t_i]$, every vertex of ${\bigcup }_{k\mathrm{\ne }i}{B}_k$ is pure to $A_j^i$.

The quantifiers range over every ambient $\bar{\mathcal{H}}$-free graph and every indicated comb, rather than fixing one graph from which a property of $\mathcal{H}$ could not follow.