Ultrafilters over uncountable cardinals and the Tukey order

We study ultrafilters on regular uncountable cardinals, with a primary focus on $\omega _1$, and particularly in relation to the Tukey order on directed sets. Results include the independence from ZFC of the assertion that every uniform ultrafilter over $\omega _1$ is Tukey-equivalent to $[2^{\aleph _1}]^{ \lt \omega }$, and for each cardinal $\kappa $ of uncountable cofinality, a new construction of a uniform ultrafilter over $\kappa $ which extends the club filter and is Tukey-equivalent to $[2^\kappa ]^{ \lt \omega }$. We also analyze Todorcevic’s ultrafilter $\mathcal {U}(T)$ under $\mathrm{PFA}$, proving that it is Tukey-equivalent to $[2^{\aleph _1}]^{ \lt \omega }$ and that it is minimal in the Rudin–Keisler order with respect to being a uniform ultrafilter over $\omega _1$. We prove that, unlike $\mathrm{PFA}$, $\mathrm{MA}_{\omega _1}$ is consistent with the existence of a coherent Aronszajn tree $T$ for which $\mathcal {U}(T)$ extends the club filter. A number of other results are obtained concerning the Tukey order on uniform ultrafilters and on uncountable directed systems.