Let ${\cal U}$ be a non-principal ultrafilter on $\omega$. If $\kappa>0$ is a cardinal, we say that a function $c:\omega \to \kappa$ is a *coloring for ${\cal U}$* if for all $U\in{\cal U}$ the restriction $c|_U$ is not constant. The *coloring number* of ${\cal U}$ is the least cardinal $\mu$ such that there is a coloring $c: \omega\to\mu$ for ${\cal U}$.

Is there a non-principal ultrafilter ${\cal U}$ on $\omega$ with finite coloring number?