Dominating numbers at singular cardinals

TL;DR

This paper investigates the generalized dominating number at singular cardinals, establishing bounds and conditions under which it differs from the continuum, using forcing and cardinal arithmetic assumptions.

Contribution

It provides new lower bounds for the dominating number at singular cardinals and clarifies when it can be less than or equal to the continuum, depending on forcing and cardinal assumptions.

Findings

In ZFC, the cofinality of the set of κ-sized subsets bounds the dominating number.

Under mild assumptions, the dominating number exceeds the size of the power set.

Forcing can make the dominating number less than the continuum under GCH and certain cofinalities.

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 $\mathfrak{d}_{\mu} = 2^{\mu}$.