Ramseyan ultrafilters (Q2759157)

The starting object is the set \((\omega)^{\leq\omega}\) of all partitions of \(\omega\). For \(A,B\in(\omega)^{\leq\omega}\), \(A\) is coarser than \(B\), if each element of \(B\) is a subset of an element of \(A\). This gives a partial order on \((\omega)^{\leq \omega}\) and makes it possible to investigate filters and ultrafilters on this set.NEWLINENEWLINENEWLINEThe author defines a partial ordering on these filters which has some similarities with the Rudin-Keisler ordering. He introduces Ramseyan ultrafilters and shows that dual Mathias forcing restricted to a Ramseyan ultrafilter has the same features as Mathias forcing restricted to a Ramsey ultrafilter. Further on the author investigates the dual form of some cardinal invariants of the continuum.