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

We separate various weak forms of Club Guessing at ω1ω1 in the presence of 2ℵ02ℵ0 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 ℵ1ℵ1-many reals but preserves CH.