Techniques for Approaching the Dual Ramsey Property in the Projective Hierarchy

Set theory of the reals is a subfield of Mathematical Logic mainly concerned with the interplay between forcing and Descriptive Set Theory. One of the motivations behind Descriptive Set Theory is the strong intuition that simple sets of real numbers should not display irregular behaviour, or, in other words, they should be topologically and measure theoretically nice. In order to fill this statement with mathematical content, we should make clear what we mean by “simple” and what we mean by “nice”. Both questions have a conventional and well known answer: • The measure of simplicity with which we categorize our sets of reals is the projective hierarchy, in other words, the number of quantifiers necessary to define the sets with a formula in first order analysis (or second order arithmetic). • A set should be considered “nice” or “regular” if it has the Baire property in all naturally occurring topologies on the real numbers and is a member of all conceivably natural σ-algebras. Set theory teaches us that the axioms of ZFC do not entail a formal version of these intuitions: It is consistent with ZFC that there are irregular sets already at the first level of the projective hierarchy.1 Thus the focus shifts from proving that all simple sets are nice to investigating the situations under which our intuitions are met by the facts.