Looking at this MO-problem, my collegue Igor Protasov suggested to ask on Mathoverflow his old question on $T$-ultrafilters hoping that somebody on MO can solve it.
First I recall the necessary definitions.
By a partition of $\omega$ we understand a cover of $\omega$ by pairwise disjoint nonempty subsets. A partition $\mathcal P$ is called finitary if $\sup_{P\in\mathcal P}|P|$ is finite.
Let $\mathfrak P$ is a family of partitions of $\omega$. An infinite subset $T\subset\omega$ is called $\mathfrak P$-thin if for any partition $\mathcal P\in\mathfrak P$ there exists a finite set $F\subset T$ such that for any $P\in\mathcal P$ the intersection $P\cap (T\setminus F)$ contains at most one point.
A free ultrafilter $\mathcal U$ on $\omega$ is called a $T$-point (or else a $T$-ultrafilter) if for any countable family $\mathfrak P$ of finitary partitions of $\omega$ there exists a $\mathfrak P$-thin set $T\in\mathcal U$.
Petrenko and Protasov proved that a free ultrafilter is a $T$-point whenever it is a $P$-point or a $Q$-point.
It is well-known that there are models of ZFC containing no $P$-points and there are models of ZFC containing no $Q$-points. But it is not known yet if there exists a model of ZFC containing neither $P$-points nor $Q$-points. That is why the following question of Protasov is interesting (and non-trivial).
Problem 1. Can the existence of $T$-points be proved in ZFC?
Problem 2. Are Kunen's OK-points $T$-points?
