Non-linear iterations and higher splitting

TL;DR

This paper investigates the preservation of certain sequences under non-linear iterations of higher Hechler forcing on strongly inaccessible cardinals, and constructs models with specific cardinal characteristics and absence of $oldsymbol{ ext{k-towers}}$.

Contribution

It demonstrates that generalized eventually narrow sequences are preserved under non-linear iterations and constructs models with prescribed cardinal invariants and no $ ext{k-towers}$.

Findings

Preservation of generalized eventually narrow sequences under non-linear iterations.

Existence of models with specified values of $ ext{cardinal invariants}$.

Construction of models with no $ ext{k-towers}$ for certain cardinals.

Abstract

We show that generalized eventually narrow sequences on a strongly inaccessible cardinal $\kappa$ are preserved under the Cummings-Shaleh non-linear iterations of the higher Hechler forcing on $\kappa$. Moreover assuming GCH, $\kappa^{<\kappa}=\kappa$, we show that: (1) if $\kappa$ is strongly unfoldable, $\kappa^+\leq\beta=\hbox{cf}(\beta)\leq \hbox{cf}(\delta)\leq\delta\leq\mu$ and $\hbox{cf}(\mu)>\kappa$,then there is a cardinal preserving generic extension in which $$\mathfrak{s}(\kappa)=\kappa^+\leq\mathfrak{b}(\kappa)=\beta\leq\mathfrak{d}(\kappa)=\delta\leq 2^\kappa=\mu.$$ (2) if $\kappa$ is strongly inaccessible, $\lambda>\kappa^+$, then in the generic extension obtained as the $<\kappa$-support iteration of $\kappa$-Hechler forcing of length $\lambda$ there are no $\kappa$-towers of length $\lambda$.