Joint work with Chris Lambie-Hanson and Jing Zhang.
Abstract. We study the interplay of the three families of combinatorial objects or principles. Specifically, we show the following.
- Strong forcing axioms, in general incompatible with the existence of indexed squares, can be made compatible with weaker versions of indexed squares.
- Indexed squares and indecomposable ultrafilters with suitable parameters can coexist. This demonstrates that the amount of stationary reflection implied by the existence of an indecomposable ultrafilter is optimal.
- The Proper Forcing Axiom implies that any cardinal carrying a uniform indecomposable ultrafilter is either measurable or a supremum of countably many measurable cardinals. Leveraging insights from the preceding sections, we demonstrate that the conclusion cannot be improved.
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