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

Given a function $f\in {\omega }^{\omega }$, a set $A\in {\left[\omega \right]}^{\omega }$ is free for f if $f\left[A\right]\cap A$ is finite. For a class of functions $\Gamma \subseteq {\omega }^{\omega }$, we define ${\mathrm{ros}}_{\Gamma }$ as the smallest size of a family $A\subseteq {\left[\omega \right]}^{\omega }$ such that for every $f\in \Gamma$ there is a set $A\in A$ which is free for f, and ${\Delta }_{\Gamma }$ as the smallest size of a family $F\subseteq \Gamma$ such that for every $A\in {\left[\omega \right]}^{\omega }$ there is $f\in F$ 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 [20] and [2].