Over 20 decades ago, Todorcevic defined a uniform filter on $\omega_1$ for each coherent Aronszajn tree $T$. He has shown that, in the presence of the Proper Forcing Axiom, this filter $\mathcal{U}(T)$ is an ultrafilter. Moreover, he showed that under these assumptions, the isomorphism type of this ultrafilter does not depend on the Aronszajn tree and that it’s projection to $\omega$ is a Ramsey ultrafilter. We extend this analysis by showing that under these assumptions, $\mathcal{U}(T)$ is on one hand minimal in the Rudin-Keisler order with respect to be uniform and on the other hand, maximal with respect to Tukey’s order on directed sets. This is joint work with Tom Benhamou and Luke Serafin.