Ultrafilters over uncountable cardinals and the Tukey order

We study ultrafilters on regular uncountable cardinals, with a primary focus on ω1, 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 ω1 is Tukey-equivalent to [2ℵ1]<ω, and for each cardinal κ of uncountable cofinality, a new construction of a uniform ultrafilter over κ which extends the club filter and is Tukey-equivalent to [2κ]<ω. We also analyze Todorcevic’s ultrafilter U(T) under PFA, proving that it is Tukey-equivalent to [2ℵ1]<ω and that it is minimal in the Rudin–Keisler order with respect to being a uniform ultrafilter over ω1. We prove that, unlike PFA, MAω1 is consistent with the existence of a coherent Aronszajn tree T for which U(T) extends the club filter. A number of other results are obtained concerning the Tukey order on uniform ultrafilters and on uncountable