The polynomial Rödl property for a finite forbidden family

Let $\mathcal{F}$ be a finite family of graphs. We say that $\mathcal{F}$ has the polynomial Rödl property if there exists a real number $d\ge 1$ such that for every real $ϵ\in (0,\frac{1}{2})$ and every nonempty $\mathcal{F}$-free finite simple graph $G$, there is an $ϵ$-restricted vertex set $X\subseteq V(G)$ with

Here $\mathcal{F}$-free is in the induced-subgraph sense of $H$-free and $\mathcal{F}$-free graphs under the induced-subgraph convention, $ϵ$-restricted means $ϵ$-sparse or $ϵ$-dense in the sense of $c$-sparse, $c$-dense and $c$-restricted vertex sets, and the power ${ϵ}^d$ is that of Real powers for positive bases, with the zero-base positive-exponent convention.

• This page keeps the maximum-degree normalization of restricted sets already fixed on the sparse-restricted-subgraphs page.
• The same exponent $d$ must work simultaneously for every $ϵ\in (0,\frac{1}{2})$ and every nonempty $\mathcal{F}$-free graph.