A small ultrafilter number at all $\aleph_3\leq\kappa<\aleph_\omega$

TL;DR

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Contribution

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Abstract

This note shows how recent work of Eskew and Hayut can be combined with a method of Raghavan and Shelah to show the consistency of $\mathfrak u_\kappa<2^\kappa$ for $\aleph_3\leq \kappa<\aleph_{\omega}$, from a huge cardinal.