Dominating numbers at singular cardinals

Abstract

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