Cardinal Exponentiation Inequality

Question

Let $\lambda, \kappa$ be infinite cardinals with $\lambda<\kappa$, what is known about $\kappa^\lambda$? specially in the case either $\kappa$ is regular. Or is there very little that can be answered about this question under ZFC (Again I'm mostly interested in the case $\kappa$ is regular)?

As you have probably guessed by now, I'm more interested in the results from ZFC with $\kappa$ regular. But I would be interested to know other results (independence results for example). I would also like to know a good source for cardinal arithmetic in general.

Edit1: Also I would like to know when $\kappa^\lambda<2^\kappa$.

Answer 1

If GCH holds, then if $\lambda < \kappa$ and $\kappa$ is regular, then $\kappa^\lambda = \kappa$.

By Easton theorem (or use product forcing), there is a model of $\mathsf{ZFC}$ such that $2^{\aleph_0} = \aleph_2$ and $2^{\aleph_1} = \aleph_3$. Then $(\aleph_2)^{\aleph_1} = (2^{\aleph_0})^{\aleph_1} = 2^{\aleph_0 \times \aleph_1} = 2^{\aleph_1} = \aleph_3$.

Answer 2

Well, here is the other side of the coin. What happens when $\kappa$ is singular? For instance, what is $\aleph_\omega^{\aleph_0}$? While this question is independent of $\mathbf{ZFC}$, we do know the an upper bound if $\aleph_\omega^{\aleph_0} \neq 2^{\aleph_0}$.

The 'result' is that when $\aleph_\omega^{\aleph_0} \neq 2^{\aleph_0}$, we have that ${\aleph_\omega^{\aleph_0}}< \aleph_{\omega_4}$.

This result was proven by Shelah and uses PCF theory. This paper has a section dedicated to trying to explain why the hell it is 4.