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 $\kappa$-Hurewicz Conjecture, and related questions on $\kappa$-Menger and $K_{\kappa }$-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 $\kappa$, we show that there is a model of ZFC in which the $\kappa$-Hurewicz Conjecture fails; in particular, it cannot hold under GCH at $\kappa$. Moreover, any model in which $\kappa$ is weakly compact and the $\kappa$-Hurewicz Conjecture holds must satisfy $2^{{\mathfrak{b}}_{\kappa }}=2^{\kappa }$. We also study the existence of ${\mathfrak{b}}_{\kappa }$-scale sets which are $K_{\kappa }$-spaces and derive a necessary tree-theoretic condition for such an example. Under the additional assumption of the Perfect Subtree Property, no ${\mathfrak{b}}_{\kappa }$-scale set is a $K_{\kappa }$-space. Consequently, under this assumption, every ${\mathfrak{b}}_{\kappa }$-scale set yields an alternative example of a $\kappa$-Menger space which is not a $K_{\kappa }$-space.