Saturated models and $\kappa=\kappa^{<\kappa}$

Question

Do not assume GCH. Can you characterize the cardinals $\kappa$ such that every theory $T$ with an infinite model has a saturated model of cardinality $\kappa$?

I guess these are the cardinals such that $\kappa=\kappa^{<\kappa}$. (Correct?) Do these cardinals have a name (are they googlable cardinals)? Do they exist in all models of ZFC? If not, how strong is their existence?

Answer 1

Unfortunately, it is consistent that if $\kappa>\aleph_0$, then $\kappa<\kappa^{<\kappa}$.

It is consistent from large cardinals (one and two) that for every infinite cardinal, $2^\kappa=\kappa^{++}$. And then we have the following:

1. If $\kappa=\lambda^+$, then $\kappa^{<\kappa}=\kappa^\lambda=\lambda^{++}=\kappa^+>\kappa$.

2. If $\kappa$ is singular, then $\lambda=\operatorname{cf}(\kappa)<\kappa$ and $\kappa^\lambda>\kappa$.

Note that the large cardinals enter the picture when blowing up the continuum of a singular cardinal, if we try to resort to $\sf ZFC$ assumptions only (read: not assuming large cardinals at all), then $\beth_{\omega+1}=(\beth_\omega)^+=\kappa$ will satisfy $\kappa^{<\kappa}=\kappa$.