Quantitative divisiveness for a finite forbidden family

A finite family $\mathcal{F}$ of finite graphs is divisive if there are $b\ge 4$ and $0<c\le 1\mathrm{/}4$ such that, for every $0<x<c$ and every nonempty finite graph $G$ with $n=\mathrm{\mid }V(G)\mathrm{\mid }\ge x^{-b\mathrm{/}2}$ and

${\mathrm{ind}}_H(G)<(x^{b}n{)}^{\mathrm{\mid }V(H)\mathrm{\mid }}(H\in \mathcal{F}),$

there are a real $2\le k\le x^{-1}$ and an $x$-sparse blockade in $G$ or an $x$-sparse blockade in $\bar{G}$, of length at least $k$ and width at least $n\mathrm{/}{k}^b$. The equivalent second alternative is a $(1-x)$-dense blockade in $G$.

The lower bound on $n$ makes every block nonempty under the existing blockade definition. It only removes the small-order case, where a singleton already satisfies the viral conclusion once its exponent is enlarged.