Let ${\cal U}$ be a non-principal [ultrafilter](https://en.wikipedia.org/wiki/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?



Coloring non-principal ultrafilters on $\omega$
If a function $c$ has finite image then $\omega$ decomposes into a disjoint union of finitely many subsets according to the value of $c$. By the property of the ultrafilter exactly one of those sets will be in $\mathcal U$. Thus $c$ is constant on a member of the ultrafilter. Therefore there is no ultrafilter with finite coloring number.


