Echeloned saturation and forcing axioms
Shimon Garti
TL;DR
The paper investigates how forcing axioms shape saturation properties of ideals on small cardinals, distinguishing Kunen, Laver, and echeloned (weakly Laver) ideals. It generalizes Larson's framework, introducing $\mathfrak{ap}_\kappa$ and polarized partition relations to analyze when saturated or dense ideals can exist. Under MA with $2^{\omega}>\aleph_2$ and failure of Chang's conjecture, there are no weakly Laver ideals on $\aleph_1$, while under Baumgartner's axiom there are no Laver ideals on $\aleph_2$ (in certain continuum configurations); these results are extended to higher cardinals under suitable hypotheses. The work clarifies the limitations of strong forcing axioms in producing echeloned saturation and raises open questions about the $\aleph_2$ case under Baumgartner-type frameworks and broader generalizations.
Abstract
Addressing a question of Paul Larson we prove the following statement. If Chang's conjecture fails, Martin's axiom holds and the continuum is greater than $\aleph_2$, there are no weakly Laver ideals over $\aleph_1$. We also prove that under Baumgartner's axiom there are no Laver ideals over $\aleph_2$.