We isolate two abstract determinacy theorems for games of length $\omega_1$ from work of Neeman and use them to conclude, from large-cardinal assumptions and an iterability hypothesis in the region of measurable Woodin cardinals that

1. (1) if the Continuum Hypothesis holds, then all games of length $\omega_1$ which are provably $\Delta_1$ -definable from a universally Baire parameter (in first-order or $\Omega $ -logic) are determined;

2. (2) all games of length $\omega_1$ with payoff constructible relative to the play are determined; and

3. (3) if the Continuum Hypothesis holds, then there is a model of ${\mathsf{ZFC}}$ containing all reals in which all games of length $\omega_1$ definable from real and ordinal parameters are determined.

We characterize the determinacy of $F_\sigma $ games of length $\omega ^2$ in terms of determinacy assertions for short games. Specifically, we show that $F_\sigma $ games of length $\omega ^2$ are determined if, and only if, there is a transitive model of ${\mathsf {KP}}+{\mathsf {AD}}$ containing $\mathbb {R}$ and reflecting $\Pi _1$ facts about the next admissible set.

As a consequence, one obtains that, over the base theory ${\mathsf {KP}} + {\mathsf {DC}} + ``\mathbb {R}$ exists,” determinacy for $F_\sigma $ games of length $\omega ^2$ is stronger than ${\mathsf {AD}}$ , but weaker than ${\mathsf {AD}} + \Sigma _1$ -separation.

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