Abstract
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.
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