  Given a function $f \in \omega^\omega$, a set $A \in [\omega]^\omega$ is free
for $f$ if $f[A] \cap A$ is finite. For a class of functions $\Gamma \subseteq
\omega^{\omega}$, we define $\mathfrak{ros}_\Gamma$ as the smallest size of a
family $\mathcal{A}\subseteq [\omega]^\omega$ such that for every $f\in\Gamma$
there is a set $A \in \mathcal{A}$ which is free for $f$, and $\Delta_\Gamma$
as the smallest size of a family $\mathcal{F}\subseteq\Gamma$ such that for
every $A\in[\omega]^\omega$ there is $f\in\mathcal{F}$ such that $A$ is not
free for $f$. We compare several versions of these cardinal invariants with
some of the classical cardinal characteristics of the continuum. Using these
notions, we partially answer some questions from arXiv:1911.01336 [math.LO] and
arXiv:2004.01979 [math.GN].
