Let $\kappa$ be an infinite cardinal. An ultrafilter ${\cal U}$ on $\kappa$ is said to be *uniform* if $|R|=\kappa$ for all $R\in{\cal U}$. If ${\cal U}$ is a non-principal ultrafilter on $\kappa$, denote by $b({\cal U})$ the minimal cardinality that a filter base for ${\cal U}$ can have.


If ${\cal U, V}$ are non-principal uniform ultrafilters on $\kappa$, do we necessarily have $b({\cal U}) = b({\cal V})$?


**Thanks** to Joseph van Name for making me aware of uniform ultrafilters and the fact that this question is only (potentially) interesting when restricted to these.



Minimal cardinality of a filter base of a non-principal uniform ultrafilters
Your number $b(U)$ is usually called the "character" of the ultrafilter $U$.


In general, there may be uniform ultrafilters on the same set with different characters. For example, it is consistent with $2^{\aleph\_0}=\aleph\_2 $ that some ultrafilters have character $\aleph\_1$, others $\aleph\_2$.


Also more complicated "character spectra" are possible, according to [Sh:915](http://shelah.logic.at/papers/915) (Topology and its Applications 158 (2011) 2535-2555)


(Of course, if $2^\kappa=\kappa^+$, then all uniform ultrafilters have
character $\kappa^+$.)


