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 . 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].

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ABSTRACT. Generalized iteration extends the usual notion of iterated forcing from iterating along an ordinal to iterating along any partially ordered set. We consider a class of forcings called perfect tree forcing. O. INTRODUCTION Generalized iterations encompass not only iterations but also product forcings and other forcing constructions which partake of the nature of each.Suppose X is any partially ordered set. An iteration along the partial order X should add to the ground model M a generic sequence, G = (G(x)lx EX) ; each G(x) should be generic over M [(G(y)ly < x)], and if Y is an initial segment of X then G ~ Y should be added by an iteration along Y. If X is an ordinal we are talking about a standard iteration, and if the ordering on X is trivial, our definition is that of a product forcing. The specific forcings of [7] and [8], designed to accomplish iterations along other partial orders, are examples of situations in which the natural forcing construction is a generalized iteration.Generalized iteration shares in the advantages and disadvantages of both product forcing and standard iteration. For example, assuming GCH in the ground model, the countable support iteration of Sacks forcing [17] along w 2 preserves cardinals [2]. The same is true if w 2 is replaced by any well-founded partial order X which is w 2 -like (i.e., for any x E X there are at most WI points below x). Suppose that X is also w 2 -directed (i.e., any size WI subset

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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.A large part of this paper consists of technical results, consideration of which was motivated by the desire to prove Theorem 8 (almost-disjoint coding via a real of minimal degree). The author would like to thank the impressive number of people who suggested this question to her (notably the first to do so, Gerald Sacks, and the last, Mack Stanley), both for the suggestion and for many helpful discussions. The final proof of this theorem owes much to conversations with Stanley. Thanks are also due to the referee for thoughtful and valuable suggestions.The title was chosen to emphasize the influence of GrigoriefT's paper, Combinatorics on ideals and forcing, [6], in suggesting the form of the questions asked here, and in providing much of the combinatorics which is used. In that paper, Grigorieff analyzes generalizations of Cohen and of Prikry-Silver forcing, essentially defined by replacing the ideal of finite sets with some larger ideal,J, on u>. He then isolates a property of J which suffices to guarantee that the generic real will be of minimal real degree over the ground model (as is a Prikry-Silver real).A common intuition in forcing is that a generic object can be forced to have some minimality property, by forcing with branching conditions. (Classic examples of this include Sacks [10] and Laver [5], [7] forcings, which add a real of minimal degree, and Prikry's forcing [3], [9] to collapse co 1 minimally.) Here we explore this idea, by performing an analysis similar to GrigoriefT's of a class of forcings whose conditions are trees on u>. The concepts formulated in [6] are used, and related results are obtained.Finally we apply these results, to give a variant of Solovay's almost-disjoint coding [8] which produces a generic real of minimal degree.The end of a proof is designated by #, the end of a definition by tl.