CEU Electronic Theses and Dissertations, 2018
| Author | Sziráki, Dorottya Ilona |
|---|---|
| Title | Colorings, Perfect Sets and Games on Generalized Baire Spaces |
| Summary | The generalized Baire space for an uncountable regular cardinal kappa (or the kappa-Baire space, for short) is the set of functions from kappa to kappa, equipped with the bounded topology. The study of the topology and descriptive set theory of these spaces is an active area of research today, with close connections to many other areas of set theory and to model theory. In this thesis we investigate, for generalized Baire spaces, the uncountable analogues of perfect set theorems and classical dichotomy theorems concerning colorings (or equivalently, graphs and hypergraphs) on the lowest levels of the Borel hierarchy. For example, we show that the kappa-analogue of the perfect set version of the Open Coloring Axiom, restricted to kappa-analytic subsets of the kappa-Baire space, is consistent relative to (and therefore equiconsistent with) the existence of an inaccessible cardinal above kappa. We also consider dichotomies for colorings on the second level of the kappa-Borel hierarchy. We prove the analogue of the previous result in the case of the kappa-Silver dichotomy for $\Sigma^0_2$ equivalence relations. We obtain perfect set theorems for homogeneous sets with respect to families of kappa many $\Pi^0_2$ colorings, some of which are strengthenings of an earlier joint result of Jouko Väänänen and the author. We also discuss different possible generalizations of the concept of perfectness for the kappa-Baire space. While these definitions of perfectness are equivalent in classical descriptive set theory, they lead to different notions in the uncountable setting. We study in detail the connections between these concepts and the games underlying (some of) their definitions. We obtain equivalent characterizations of several dichotomies studied in this thesis in terms of these perfect set games and their natural analogues for open colorings. We also consider generalizations of density for the kappa-Baire space which correspond to the different notions of perfectness, and look at a "dense in itself subset property" for the kappa-Baire space. The approximations (in the sense of [Hyttinen1990]) of the games discussed here can be used to generalize Cantor-Bendixson ranks for subsets of the kappa-Baire space [Väänänen1991] and for subtrees of $\kappa^{<\kappa}$, as well as different analogues of these ranks for open colorings. We prove results on the connections between these games which can be interpreted as comparisons of the levels of the different generalized hierarchies associated to these games. |
| Supervisor | Sági, Gábor |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2018/sziraki_dorottya.pdf |
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