Distributivity and minimality in perfect tree forcings for singular cardinals

Dobrinen, Hathaway and Prikry studied a forcing ℙ κ consisting of perfect trees of height λ and width κ where κ is a singular λ-strong limit of cofinality λ. They showed that if κ is singular of countable cofinality, then ℙ κ is minimal for ω -sequences assuming that κ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption. Prikry proved that ℙ κ is ( ω , ν )-distributive for all ν < κ given a singular ω -strong limit cardinal κ of countable cofinality, and Dobrinen et al. asked whether this result generalizes if κ has uncountable cofinality. We answer their question in the negative by showing that ℙ κ is not (λ, 2)-distributive if κ is a λ-strong limit of uncountable cofinality λ 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 ℙ κ in particular is not ( ω , ·, λ + )-distributive under these assumptions. While developing these ideas, we address natural questions regarding minimality and collapses of cardinals.