For an infinite cardinal κ, the ultrafilter number u_κ is the smallest size of a family that generates a uniform ultrafilter on κ. We say that u_κ is bounded if u_κ<2^κ.

The case κ=ω is well understood, with the consistency of a bounded ultrafilter number at $\omega$ first noted in the early 1970s by Kunen. The method used in Kunen’s construction does not generalize to the case κ=ω_1, and Kunen asked whether u_{ω_1}<2^{ω_1} is even consistent; this remains wide open.

The consistency of u_κ<2^κ for general uncountable κ has been studied extensively in the last decade. In this series of talks, we will survey some recent progress towards bounding the ultrafilter number at successor values of κ, with emphasis on a key theorem of Raghavan and Shelah. We will also discuss some crucial limitations to applying this theorem for a bounded ultrafilter number at any of the ω_n’s.