$F_\sigma $ GAMES AND REFLECTION IN $L(\mathbb {R})$. (21st September 2020) - AGUILERA, J. P.
Abstract: 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.