On Ramsey Property under the Axiom of Determinacy

Abstract

The work of this thesis is motivated by the open problem whether the Axiom of Determinacy implies every set of reals is Ramsey.

First, we reduce the open problem to a problem for sets with some certain property. We define two reals to be equivalent if they differ only at a finite part, and a set of reals to be invariant if it is a union of some equivalent classes. We proposed the weakly Ramsey property, which is a connection between the Ramsey property and invariant sets. By some analysis on the behavior of weakly Ramsey sets, it is proved in this thesis that if every invariant set is Ramsey then every set is Ramsey in the context of ZF+DC+AD.

Second, It is reasonable to run an induction on the Wadge rank. And we did some investigation into the Wadge rank of invariant sets. It is summarized by Theorem 4.2.3.