We show that there may be a Milliken-Taylor ultrafilter with infinitely many near coherence classes of ultrafilters in its projection to ω, answering a question by López-Abad. We show that k-colored Milliken-Taylor ultrafilters have at least k+1 near coherence classes of ultrafilters in its projection to ω. We show that the Mathias forcing with a Milliken-Taylor ultrafilter destroys all Milliken-Taylor ultrafilters from the ground model.
"On Milliken-Taylor Ultrafilters." Notre Dame J. Formal Logic 52 (4) 381 - 394, 2011. https://doi.org/10.1215/00294527-1499345