Let $\text{NPU}(\omega)$ be the set of non-principal [ultafilters][1] on $\omega$. The *Rudin-Keisler preorder* on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\exists f:\omega\to\omega)(\forall U\in{\cal U}) f^{-1}(U)\in {\cal V} .$$

It is easy to see that $\leq_{RK}$ is reflexive and transitive, but not anti-symmetric. Set ${\cal U}\simeq_{RK} {\cal V}$ if ${\cal U}\leq_{RK}{\cal V}$ and ${\cal V}\leq_{RK}{\cal U}$. So $\text{NPU}(\omega)/\simeq_{RK}$ is a poset with the Rudin-Keisler order applied to equivalence classes.

It is known that ${\cal U}$ is a minimal element of $\text{NPU}(\omega)/\simeq_{RK}$ if and only if ${\cal U}$ is a Ramsey ultrafilter. (Ramsey ultrafilters do not necessarily exist.)

**Question.** Does $\text{NPU}(\omega)/\simeq_{RK}$ have maximal elements?