Dominating numbers at singular cardinals

Abstract

We study the generalized dominating number ${\mathfrak{d}}_{\hat{I}}¼$ at a singular cardinal $\hat{I}¼$ of cofinality $\hat{I}º$. We show two lower bounds: in ZFC, $\mathrm{cf}([\hat{I}¼{]}^{\hat{I}}º,\subseteq )\le {\mathfrak{d}}_{\hat{I}}¼$, and under mild cardinal-arithmetic assumptions, $2^{<\hat{I}¼}\le {\mathfrak{d}}_{\hat{I}}¼$. We also clarify when ${\mathfrak{d}}_{\hat{I}}¼$ can differ from $2^{\hat{I}}¼$: assuming GCH and $\hat{I}º=\mathrm{cf}(\hat{I}¼)>\ddot{I}‰$, a finite-support iteration of Cohen forcing of length $\hat{I}{¼}^{++}$ yields ${\mathfrak{d}}_{\hat{I}}¼<2^{\hat{I}}¼$. On the other hand, for $\hat{I}º=\mathrm{cf}(\hat{I}¼)=\ddot{I}‰$, natural $\hat{I}¼$-cc posets force ${\mathfrak{d}}_{\hat{I}}¼=2^{\hat{I}}¼$.