For an uncountable cardinal $\kappa$, we are interested in the least size of a cofinal subset of the partial order $([\kappa]^\omega, \subseteq)$. It is obvious that this cofinality is at least $\kappa$ and a rather simple induction shows that $cof([\aleph_n]^\omega, \subseteq)=\aleph_n$ for each natural number $n \geq 1$.
What is known in $ZFC$ for bigger cardinals?
Also I am intrigued by the following statement made by Alan Dow in Efimov spaces and the splitting number, Topology Proc. (2005):
It is a "large cardinal" hypothesis to assume that there is a cardinal $\kappa$ with uncountable cofinality such that this cofinality is greater than $\kappa$.
What exactly does he mean by that, and what is a good reference to read about it?