Forcing Many Positive Polarized Partition Relations Between A Cardinal and Its Powerset

A fairly quotable special, but still representative, case of our main result is that for 2 ≤ n < ω, there is a natural numberm(n) such that, the following holds. Assume GCH: If λ < μ are regular, there is a cofinality preserving forcing extension in which 2 = μ and, for all σ < λ ≤ κ < η such that η(+m(n)−1) ≤ μ, ((η)σ)→ ((κ)σ) (1)n η . This generalizes results of [3], Section 1, and the forcing is a “many cardinals” version of the forcing there. §0. INTRODUCTION. In [3], the first author proved (with, in what follows, μ in the place of our λ, and λ in the place of our η) the consistency of: λ < κ < η are all regular, 2 = η, η → (η, [κ;κ]) The forcing can be thought of as a “filtering through” κ of adding η many Cohen subsets of λ. Then, {λ, κ, η} can be thought of as a three element set K of regular cardinals used for defining the forcing; the elements of K are taken, in the ground model, to be sufficiently far apart. An important technical notion, related to the idea of “filtering through” is the possibility of viewing p ≤ q as split up, in various ways, into “pure” and “apure” extensions. Schematically, but fairly accurately, the pure extensions have completeness properties, while the apure extensions have chain condition properties: see (1.7) for the former and (1.8), (1.9) for the latter. It is natural to attempt to allow the set K of regular cardinals to be larger, and to simultaneously obtain many such, and stronger, partition relations, for example, by increasing the “dimension” (from 2 to n) and the The research of the first author was partially supported by the NSF and the Basic Research Fund, Israel Academy of Science. This is paper number 608 in the first author’s list of publications. We thank the referee for many helpful suggestions.