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 $\mathrm{\Gamma }\subseteq {\omega }^{\omega }$, we define ${\mathfrak{r}\mathfrak{o}\mathfrak{s}}_{\mathrm{\Gamma }}$ as the smallest size of a family $\mathcal{A}\subseteq [\omega {]}^{\omega }$ such that for every $f\in \mathrm{\Gamma }$ there is a set $A\in \mathcal{A}$ which is free for $f$, and ${\mathrm{\Delta }}_{\mathrm{\Gamma }}$ as the smallest size of a family $\mathcal{F}\subseteq \mathrm{\Gamma }$ 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].