The viral property for a finite forbidden family

Let $\mathcal{F}$ be a finite family of finite graphs. We say that $\mathcal{F}$ is viral if there exists a real number $d\ge 1$ such that for every real $ϵ\in (0,\frac{1}{2})$ and every nonempty finite simple graph $G$ satisfying

${\mathrm{ind}}_H(G)<({ϵ}^d\mathrm{\mid }V(G)\mathrm{\mid }{)}^{\mathrm{\mid }V(H)\mathrm{\mid }}\text{for every }H\in \mathcal{F},$

there is an $ϵ$-restricted vertex set $X\subseteq V(G)$ with

$\mathrm{\mid }X\mathrm{\mid }\ge {ϵ}^d\mathrm{\mid }V(G)\mathrm{\mid }\mathrm{.}$

The induced-copy count is that of The induced-embedding count ${\mathrm{ind}}_H(G)$, and $ϵ$-restricted is in the sense of $c$-sparse, $c$-dense and $c$-restricted vertex sets.

• Viral families are allowed to be vacuous: if the displayed copy bounds have no nonempty instances, the implication is still true.
• The source formulation is already in the induced-copy and restricted-set language used on this page, so no change of normalization is hidden here.