In this article we give a forcing characterization for the Ramsey property of Σ 1 2 -sets of reals. This research was motivated by the well-known forcing characterizations for Lebesgue measurability and the Baire property of Σ 1 2 -sets of reals. Further we will show the relationship between higher degrees of forcing absoluteness and the Ramsey property of projective sets of reals.

We explore the connection between combinatorial principles on uncountable cardinals, like stick and club, on the one hand, and the combinatorics of sets of reals and, in particular, cardinal invariants of the continuum, on the other hand. For example, we prove that additivity of measure implies that Martin’s axiom holds for any Cohen algebra. We construct a model in which club holds, yet the covering number of the null ideal is large. We show that for uncountable cardinals κ≤λ and , (...) if all subsets of λ either contain, or are disjoint from, a member of , then has size at least etc. As an application, we solve the Gross space problem under by showing that there is such a space over any countable field. In two appendices, we solve problems of Fuchino, Shelah and Soukup, and of Kraszewski, respectively. (shrink)