Geodesic
Squares, ultrafilters and forcing axioms
Zbornik radova, Tome 22 (2025) no. 30, p. 411

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DOI

We study relationships between various set-theoretic compactness principles, focusing on the interplay between the three families of combinatorial objects or principles mentioned in the title. Specifically, we show the following. (1) Strong forcing axioms, in general incompatible with the existence of indexed squares, can be made compatible with weaker versions of indexed squares. (2) Indexed squares and indecomposable ultrafilters with suitable parameters can coexist. This demonstrates that the amount of stationary reflection known to be implied by the existence of a uniform indecomposable ultrafilter is optimal. (3) 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.
DOI : 10.64191/zr24040410112
Classification : 03E35, 03E57
Keywords: proper forcing axiom, Martin's maximum, semi-sta\-tionary reflection principle, narrow system property, guessing models, indexed square, indecomposable ultrafilters, finest partition, C-se-quence number, ascent path 
Chris Lambie-Hanson; Assaf Rinot; Jing Zhang. Squares, ultrafilters and forcing axioms. Zbornik radova, Tome 22 (2025) no. 30, p. 411 . doi: 10.64191/zr24040410112
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