# Assuming CH, can every tower be extended to a selective ultrafilter (or even a p-point)?

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.)
• For the existence of non-Ramsey ultrafilters you can also observe that if $p\in\omega^*$, then $p\cdot p$ is never Ramsey, where $\cdot$ is the Fubini (tensor) product. – Brian M. Scott Jan 18 '16 at 22:32
• I should point out that Noah Schweber's argument is incorrect: Under CH, an ultrafilter $\mathcal{U}$ is generated by a tower if and only if $\mathcal{U}$ is a p point. There are ultrafilters which are not p points under CH. – Horse Jan 21 '16 at 1:23