Complete Combinatorics For Ultrafilters And The Rudin-keisler Ordering.
Claude Laflamme · Deep Blue (University of Michigan) · 1987
Mathias has shown that forcing with ${\rm I\!P}$ = $$ yields a Ramsey ultrafilter on the natural numbers; more significantly it easily follows from his work that being Ramsey constitutes complete combinatorics for ${\rm I\!P}$, i.e. any Ramsey ultrafilter in the model V (G) obtained by Levy-collapsing a Mahlo cardinal to $\aleph\sb 1$ is ${\rm I\!P}$-generic over ${\rm HOD(I\!R})\sp{{\bf V}\lbrack{\rm G}\rbrack}.$ We generalize the method and show for example how to obtain via a forcing notion ${\rm I\!P}\sb n$ a semiselective ultrafilter U satisfying $\omega\to$ (U) $\sbsp{n\sp{k-1}+1}{k}$ for all $k\in\omega$, and that these combinatorial properties are complete for ${\rm I\!P}\sb n$ in the above sense. We use one of our forcing notions to solve a problem of Baumgartner and Taylor (1978): we produce via forcing ultrafilters U,V such that $U\otimes V$ is arrow but $V\otimes U$ is not. As another application, we produce for any truth value not outright inconsistent that we can assign to being rapid, P-point or Q-point, via forcing an ultrafilter all of whose Rudin-Keisler predecessors satisfy the particular truth assignment. Finally, we answer a question of Blass (1973): we show that ZFC does not decide whether two P-points which have a common RK lower bound necessarily have a common RK upper bound which is a P-point.