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?
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