Partition theorems from creatures and idempotent ultrafilters (Q360354)

In previous papers, e.g., [Mem. Am. Math. Soc. 671, 167 p. (1999; Zbl 0940.03059)], the authors have developed a very general and flexible framework for the construction of proper forcings that add reals of one kind without adding reals of other kinds. In this paper, the authors isolate a combinatorial common denominator of the properness arguments. The main idea is to create the desired generic condition from homogeneous sets for partitions and these homogeneous sets are in turn obtained using idempotent ultrafilters with respect to various operations on the Čech-Stone remainder of~\(\omega\). Unfortunately, the machinery takes a lot of definitions to set up, too many to summarize adequately.