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      Some remarks on uncountable rainbow Ramsey theory

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          Abstract

          We discuss the rainbow Ramsey theorems at inaccessible cardinals and successors of singular cardinals, answering some questions in \cite{MR2354904} and \cite{MR2902230}. We also demonstrate extents of possible generalizations to nonspecial trees and partition relations of higher exponents. Finally, we show the coloring in the model constructed in \cite{MR2902230} is indestructible under strongly proper forcings but destructible under some c.c.c forcing.

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          Saturated ideals

          Kenneth Kunen (1978)
          In this paper, we give consistency proofs for the existence of a κ-saturated ideal on an inaccessible κ, and for the existence of an ω2-saturated ideal on ω1. We also include an historical survey outlining other known results on saturated ideals.
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            A very weak square principle

            Matthew Foreman, Menachem Magidor (1997)
            In this paper we explicate a very weak version of the principle □ discovered by Jensen who proved it holds in the constructible universe L. This principle is strong enough to include many of the known applications of □, but weak enough that it is consistent with the existence of very large cardinals. In this section we show that this principle is equivalent to a common combinatorial device, which we call a Jensen matrix. In the second section we show that our principle is consistent with a supercompact cardinal. In the third section of this paper we show that this principle is exactly equivalent to the statement that every torsion free Abelian group has a filtration into σ-balanced subgroups. In the fourth section of this paper we show that this principle fails if you assume the Chang's Conjecture: In the fifth section of the paper we review the proofs that the various weak squares we consider are strictly decreasing in strength. Section 6 was added in an ad hoc manner after the rest of the paper was written, because the subject matter of Theorem 6.1 fit well with the rest of the paper. It deals with a principle dubbed “Not So Very Weak Square”, which appears close to Very Weak Square but turns out not to be equivalent.
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              Ideals and Generic Elementary Embeddings

              Matthew D. Foreman (2010)
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                Publication date Created: 03 September 2018
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                ArXiV ID: 1809.00649
                SO-VID: 7633bc46-232e-43df-a0c1-25736ad97dff
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                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                ScienceOpen disciplines: Logic & Foundation
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                ScienceOpen disciplines: Logic & Foundation

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