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

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### Abstract

We separate various weak forms of Club Guessing at $$\omega_1$$ in the presence of $$2^{\aleph_0}$$ 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 $$\omega$$-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 $$\aleph_1$$-many reals but preserves CH.

### Details

Original language English 284–308 Annals of Pure and Applied Logic 167 3 21 Dec 2015 https://doi.org/10.1016/j.apal.2015.12.003 Published - Mar 2016 Yes

### Research areas

• Iterated forcing, Club-guessing principles, Side conditions, Polychromatic Ramsey theory

ID: 31262424

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