We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using this method we prove that the forcing axiom for the class of all finitely proper posets of size ℵ 1 is compatible with 2 ℵ 0 > ℵ 2 . In particular, this answers a question of Moore by showing that does not follow from this arithmetical assumption.

Abstract. We define the ℵ 1.5 -chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom; in fact, MA 1.5 <κ implies MA <κ . Also, MA 1.5 <κ implies certain uniform failures of club-guessing on ω 1 that do not seem to have been considered in the literature before. We show, assuming CH and given any regular cardinal κ ≥ ω 2 such that µ ℵ0 < κ for all µ < κ and such that ♦({α < κ : cf(α) ≥ ω 1 }) holds, that there is a proper ℵ 2 -c.c. partial order of size κ forcing 2 ℵ0 = κ together with MA 1.5 <κ .1. A generalization of Martin's Axiom. And some of its applications.Martin's Axiom, often denoted by MA, is the following very wellknown and very classical forcing axiom: If P is a partial order (poset, for short) with the countable chain condition 1 and D is a collection of size less than 2 ℵ 0 consisting of dense subsets of P, then there is a filterMartin's Axiom is obviously a weakening of the Continuum Hypothesis. Given a cardinal λ, MA λ is obtained from considering, in the above formulation of MA, collections D of size at most λ rather than of size less than 2 ℵ 0 . Martin's Axiom becomes interesting when 2 ℵ 0 > ℵ 1 . MA ℵ 1 was the first forcing axiom ever considered ([9]). As observed by D. Martin, the consistency of MA together with 2 ℵ 0 > ℵ 1 follows from generalizing the Solovay-Tennenbaum construction of a model of Suslin's Hypothesis by iterated forcing using finite supports ([14]). Since then, a plethora of applications of MA (+ 2 ℵ 0 > ℵ 1 ) have been

Abstract. Measuring says that for every sequence (C δ ) δ<ω1 with each C δ being a closed subset of δ there is a club C ⊆ ω 1 such that for every δ ∈ C, a tail of C ∩ δ is either contained in or disjoint from C δ . We answer a question of Justin Moore by building a forcing extension satisfying measuring together with 2 ℵ0 > ℵ 2 . The construction works over any model of ZFC + CH and can be described as a finite support forcing iteration with systems of countable models as side conditions and with symmetry constraints imposed on its initial segments. One interesting feature of this iteration is that it adds dominating functions f : ω 1 −→ ω 1 mod. countable at each of its stages.

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

Abstract. We develop a general framework for forcing with coherent adequate sets on H(λ) as side conditions, where λ ≥ ω 2 is a cardinal of uncountable cofinality. We describe a class of forcing posets which we call coherent adequate type forcings. The main theorem of the paper is that any coherent adequate type forcing preserves CH. We show that there exists a forcing poset for adding a club subset of ω 2 with finite conditions while preserving CH, solving a problem of Friedman [3].