Selection principles and the Scheepers diagram in an uncountable setting

Abstract

We continue the study of selection principles in the uncountable setting initiated in [18] and further developed in [1]. We investigate several problems posed in [1] concerning the Scheepers diagram, the $κ$-Hurewicz Conjecture, and related questions on $κ$-Menger and $K_κ$-spaces. We determine the Scheepers diagram in the uncountable setting up to three unresolved implications and compute the critical cardinalities of several properties occurring in the diagram. For weakly compact $κ$, we show that there is a model of ZFC in which the $κ$-Hurewicz Conjecture fails; in particular, it cannot hold under GCH at $κ$. Moreover, any model in which $κ$ is weakly compact and the $κ$-Hurewicz Conjecture holds must satisfy $2^{\mathfrak{b}_κ} = 2^κ$. We also study the existence of $\mathfrak{b}_κ$-scale sets which are $K_κ$-spaces and derive a necessary tree-theoretic condition for such an example. Under the additional assumption of the Perfect Subtree Property, no $\mathfrak{b}_κ$-scale set is a $K_κ$-space. Consequently, under this assumption, every $\mathfrak{b}_κ$-scale set yields an alternative example of a $κ$-Menger space which is not a $K_κ$-space.