Small Cardinals and the Pseudocompactness of Hyperspaces of Subspaces of $\beta \omega$

We study the relations between a generalization of pseudocompactness, named $(\kappa ,M)$-pseudocompactness, the countably compactness of subspaces of $\beta \omega$ and the pseudocompactness of their hyperspaces. We show, by assuming the existence of $\mathfrak{c}$-many selective ultrafilters, that there exists a subspace of $\beta \omega$ that is $(\kappa ,{\omega }^{\ast })$-pseudocompact for all $\kappa <\mathfrak{c}$, but $\text{CL}(X)$ isn’t pseudocompact. We prove in ZFC that if $\omega \subseteq X\subseteq \beta \omega$ is such that $X$ is $(\mathfrak{c},{\omega }^{\ast })$-pseudocompact, then $\text{CL}(X)$ is pseudocompact, and we further explore this relation by replacing $\mathfrak{c}$ for some small cardinals. We provide an example of a subspace of $\beta \omega$ for which all powers below $\mathfrak{h}$ are countably compact whose hyperspace is not pseudocompact, we show that if $\omega \subseteq X$, the pseudocompactness of $\text{CL}(X)$ implies that $X$ is $(\kappa ,{\omega }^{\ast })$-pseudocompact for all $\kappa <\mathfrak{h}$, and provide an example of such an $X$ that is not $(\mathfrak{b},{\omega }^{\ast })$-pseudocompact.