Assume CH. A tower is an almost decreasing family $(A_\alpha)_{\alpha\in\omega_1}$ with no pseudointersection. A selective (also called Ramsey) ultrafilter is one with the property that for every $f:[\omega]^{2}\rightarrow2$, there is some set $X$ in the ultrafilter with $|f''(X)|=1$.
As in the title, my question is whether or not every tower can be extended to a selective ultrafilter?
The answer is no: for every ultrafilter $U$, there is a tower $T$ such that the only ultrafilter containing $T$ is $U$. (Not every tower has this property, of course.)
So it remains to show that there exist non-Ramsey ultrafilters, which we can do by diagonalizing against all possible colorings.