= Ultrafilter with arithmetic-progression-rich members
{title2=$\mathcal U\subseteq\mathrm{AP}$}
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>.
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