Separating club-guessing principles in the presence of fat forcing axioms

We separate various weak forms of Club Guessing at ω1 in the presence of 2ℵ0 large, Martin's Axiom, and related forcing axioms. We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum lar...

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Bibliographic Details
Published inAnnals of pure and applied logic Vol. 167; no. 3; pp. 284 - 308
Main Authors Asperó, David, Mota, Miguel Angel
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.03.2016
Subjects
Club-guessing principles
Iterated forcing
Polychromatic Ramsey theory
Side conditions
03E50
Iterated forcing
03E57
03E35
Club-guessing principles
Polychromatic Ramsey theory
03E05
Side conditions
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Summary:We separate various weak forms of Club Guessing at ω1 in the presence of 2ℵ0 large, Martin's Axiom, and related forcing axioms. We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum large. All these models are generic extensions via finite support iterations with symmetric systems of structures as side conditions, possibly enhanced with ω-sequences of predicates, and in which the iterands are taken from a relatively small class of forcing notions. We also prove that the natural forcing for adding a large symmetric system of structures (the first member in all our iterations) adds ℵ1-many reals but preserves CH.
ISSN:0168-0072
DOI:10.1016/j.apal.2015.12.003