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

Abstract We separate various weak forms of Club Guessing at ω 1 in the presence of 2 ℵ 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 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 ℵ 1 -many reals but preserves CH.