Dominating numbers at singular cardinals

Abstract

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