On cardinal invariants related to Rosenthal families and large-scale topology

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].