We show, relative to the base theory RCA₀: A nontrivial tree satisfies Ramsey's Theorem only if it is biembeddable with the complete binary tree. There is a class of partial orderings for which Ramsey's Theorem for pairs is equivalent to ACA₀. Ramsey's Theorem for singletons for the complete binary tree is stronger than $B\sum_{2}^{0}$ , hence stronger than Ramsey's Theorem for singletons for ω. These results lead to extensions of results, or answers to questions, of Chubb, Hirst, and McNicholl [3].

Uncountable superperfect forcing is tree forcing on regular uncountable cardinals κ with κ<κ=κ, using trees in which the heights of nodes that split along any branch in the tree form a club set, and such that any node in the tree with more than one immediate extension has measure-one-many extensions, where the measure is relative to some κ-complete, nonprincipal normal filter F. This forcing adds a generic of minimal degree if and only if F is κ-saturated.

Classes of forcings which add a real by forcing with branching conditions are examined, and conditions are found which guarantee that the generic real is of minimal degree over the ground model. An application is made to almost-disjoint coding via a real of minimal degree.

We show that if there is a nonconstructible real, then every perfect set has a nonconstructible element, answering a question of K. Prikry. This is a specific instance of a more general theorem giving a sufficient condition on a pair $M\subset N$ of models of set theory implying that every perfect set in N has an element in N which is not in M.

The Sacks Density Theorem [7] states that the Turing degrees of the recursively enumerable sets are dense. We show that the Density Theorem holds in every model of P - + BΣ 2 . The proof has two components: a lemma that in any model of P - + BΣ 2 , if B is recursively enumerable and incomplete then IΣ 1 holds relative to B and an adaptation of Shore's [9] blocking technique in α-recursion theory to models of arithmetic.

Grigorieff showed that forcing to add a subset of ω using partial functions with suitably chosen domains can add a generic real of minimal degree. We show that forcing with partial functions to add a subset of an uncountable κ without adding a real never adds a generic of minimal degree. This is in contrast to forcing using branching conditions, as shown by Brown and Groszek.

Theorem. There is a non-empty Π10 class of reals, each of which computes a real of minimal degree. Corollary. WKL “there is a minimal Turing degree”. This answers a question of H. Friedman and S. Simpson.

We answer several questions posed by Hamkins and Leahy concerning the implicitly constructible universe Imp, which they introduced in [5]. Specifically, we show that it is relatively consistent with ZFC that $$Imp = \neg {\rm{CH}}$$, that $Imp \ne {\rm{HOD}}$, and that $$Imp \models V \ne Imp$$, or in other words, that $\left^{Imp} \ne Imp$.