Perfect subsets of generalized Baire spaces and long games

We extend Solovay's theorem about definable subsets of the Baire space to the generalized Baire space ${}^\lambda\lambda$, where $\lambda$ is an uncountable cardinal with $\lambda^{<\lambda}=\lambda$. In the first main theorem, we show that that the perfect set property for all subsets of ${}^{\lambda}\lambda$ that are definable from elements of ${}^\lambda\mathrm{Ord}$ is consistent relative to the existence of an inaccessible cardinal above $\lambda$. In the second main theorem, we introduce a Banach-Mazur type game of length $\lambda$ and show that the determinacy of this game, for all subsets of ${}^\lambda\lambda$ that are definable from elements of ${}^\lambda\mathrm{Ord}$ as winning conditions, is consistent relative to the existence of an inaccessible cardinal above $\lambda$. We further obtain some related results about definable functions on ${}^\lambda\lambda$ and consequences of resurrection axioms for definable subsets of ${}^\lambda\lambda$

Laver trees in the generalized Baire space

We prove that any suitable generalization of Laver forcing to the space κκ, for uncountable regular κ, necessarily adds a Cohen κ-real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. Using this dichotomy, we prove the following stronger result: if κ&lt;κ = κ, then every &lt;κ-distributive tree forcing on κκ adding a dominating κ-real which is the image of the generic under a continuous function in the ground model, adds a Cohen κ-real. This is a contribution to the study of generalized Baire spaces and answers a question from [1]

The open dihypergraph dichotomy for generalized Baire spaces and its applications

The open graph dichotomy for a subset $X$ of the Baire space ${}^\omega\omega$ states that any open graph on $X$ either admits a coloring in countably many colors or contains a perfect complete subgraph. This strong version of the open graph axiom for $X$ was introduced by Feng and Todor\v{c}evi\'c to investigate definable sets of reals. We first show that its recent generalization to infinite dimensional directed hypergraphs by Carroy, Miller and Soukup holds for all subsets of the Baire space in Solovay's model, extending a theorem of Feng in dimension $2$. The main theorem lifts this result to generalized Baire spaces ${}^\kappa\kappa$ in two ways. (1) For any regular infinite cardinal $\kappa$, the following holds after a L\'evy collapse of an inaccessible cardinal $\lambda>\kappa$ to $\kappa^+$. Suppose that $H$ is a $\kappa$-dimensional box-open directed hypergraph on a subset of ${}^\kappa\kappa$ such that $H$ is definable from a $\kappa$-sequence of ordinals. Then either $H$ admits a coloring in $\kappa$ many colors or there exists a continuous homomorphism from a canonical large directed hypergraph to $H$. (2) If $\lambda$ is a Mahlo cardinal, then the previous result extends to all box-open directed hypergraphs on any subset of ${}^\kappa\kappa$ that is definable from a $\kappa$-sequence of ordinals. We derive several applications to definable subsets of generalized Baire spaces, among them variants of the Hurewicz dichotomy that characterizes subsets of $K_\sigma$ sets, an asymmetric version of the Baire property, an analogue of the Kechris-Louveau-Woodin dichotomy that characterizes when two disjoint sets can be separated by an $F_\sigma$ set, the determinacy of V\"a\"an\"anen's perfect set game for all subsets of ${}^\kappa\kappa$, and an analogue of the Jayne-Rogers theorem that characterizes functions which are $\sigma$-continuous with closed pieces.Comment: 115 pages, 11 figures. Added new results in Section 6.2.2 which strengthen and replace the results in Section 6.3 of the previous version. Improved results in Section 5.3. Various other minor corrections. Comments are welcom