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?**



What is the cofinality of $([\kappa]^\omega, \subseteq)$?
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.](http://math.ipm.ac.ir/~golshani/Papers/On%20a%20question%20of%20Gorelic.pdf)


