Ultrafilter with arithmetic-progression-rich members ($\mathcal{U}\subseteq \mathrm{AP}$)

An ultrafilter on the positive integers with every member containing finite arithmetic progressions of arbitrarily large length. The Van der Waerden theorem and ultrafilter selection from a partition-rich family give existence. Such an ultrafilter must contain the complement of every set with bounded arithmetic-progression length, and it is a nonprincipal ultrafilter. No corresponding ultrafilter can have an infinite arithmetic progression in every member, by the dyadic block obstruction to infinite arithmetic progressions.