Distributivity and Minimality in Perfect Tree Forcings for Singular Cardinals

Dobrinen, Hathaway and Prikry studied a forcing $\mathbb{P}_\kappa$ consisting of perfect trees of height $\lambda$ and width $\kappa$ where $\kappa$ is a singular $\omega$-strong limit of cofinality $\lambda$. They showed that if $\kappa$ is singular of countable cofinality, then $\mathbb{P}_\kappa$ is minimal for $\omega$-sequences assuming that $\kappa$ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption. Prikry proved that $\mathbb{P}_\kappa$ is $(\omega,\nu)$-distributive for all $\nu<\kappa$ given a singular $\omega$-strong limit cardinal $\kappa$ of countable cofinality, and Dobrinen et al$.$ asked whether this result generalizes if $\kappa$ has uncountable cofinality. We answer their question in the negative by showing that $\mathbb{P}_\kappa$ is not $(\lambda,2)$-distributive if $\kappa$ is a $\lambda$-strong limit of uncountable cofinality $\lambda$ and we obtain the same result for a range of similar forcings, including one that Dobrinen et al$.$ consider that consists of pre-perfect trees. We also show that $\mathbb{P}_\kappa$ in particular is not $(\omega,\cdot,\lambda^+)$-distributive under these assumptions. While developing these ideas, we address natural questions regarding minimality and collapses of cardinals.