TITLE: What is the cofinality of $([\kappa]^\omega, \subseteq)$?
QUESTION [10 upvotes]: 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?

REPLY [10 votes]: I think he means the consistency of "there exists a cardinal $\kappa$ with uncountable cofinality so that $cf([\kappa]^\omega, \subseteq) > \kappa$" requires large cardinal axioms.
To motivate this, note that for example if $0^\sharp$ does not exist, then  for any cardinal $\kappa > 2^{\aleph_0}$ of uncountable cofinality we have $cf([\kappa]^\omega, \subseteq) = \kappa$.
In fact one can say more. A related argument is given in Theorem 3.1 of   Singular cofinality conjecture and a question of Gorelic.