Monotonicity of the ultrafilter number function

TL;DR

This paper investigates the monotonicity of the ultrafilter number function $\kappa \mapsto \mathfrak{u}(\kappa)$ across cardinals. It shows that monotonicity can fail in models with strong large-cardinal assumptions, while also proving numerous ZFC restrictions that constrain where such failures can occur, including positive results like $\mathfrak{u}(\kappa) \le \mathfrak{u}(\kappa^+)$ for certain singulars. The authors employ Raghavan–Shelah forcing, indecomposable ultrafilters, and PCF/square techniques to both construct failures and derive structural restrictions, including equiconsistency results with measurable cardinals and lower bounds involving strong cardinals. The work thus clarifies the delicate interaction between ultrafilter bases, indecomposability, and large-cardinal hypotheses, and raises open questions about the exact strength needed for broader failures.

Abstract

We investigate whether the ultrafilter number function $κ\mapsto \mathfrak{u}(κ)$ on the cardinals is monotone, that is, whether $\mathfrak{u}(λ) \le \mathfrak{u}(κ)$ holds for all cardinals $λ< κ$ or not. We show that monotonicity can fail, but the failure has large cardinal strength. On the other hand, we prove that there are many restrictions of the failure of monotonicity. For instance, if $κ$ is a singular cardinal with countable cofinality or a strong limit singular cardinal, then $\mathfrak{u}(κ) \le \mathfrak{u}(κ^+)$ holds.

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