Definable combinatorics at the first uncountable cardinal

Abstract:

We work throughout in the theory $\mathsf {ZF}$ with the axiom of determinacy, $\mathsf {AD}$. We introduce and prove some club uniformization principles under $\mathsf {AD}$ and $\mathsf {AD}_\mathbb {R}$. Using these principles, we establish continuity results for functions of the form $\Phi \colon [{\omega _{1}}]^{\omega _{1}} \rightarrow {\omega _{1}}$ and $\Psi \colon [{\omega _{1}}]^{\omega _{1}} \rightarrow {}^{\omega _{1}}{\omega _{1}}$. Specifically, for every function $\Phi \colon [\omega _1]^{\omega _1} \rightarrow \omega _1$, there is a club $C \subseteq \omega _1$ so that $\Phi \upharpoonright [C]^{\omega _1}_*$ is a continuous function. This has several consequences such as establishing the cardinal relation $|[{\omega _{1}}]^{<{\omega _{1}}}| < |[{\omega _{1}}]^{\omega _{1}}|$ and answering a question of Zapletal by showing that if $\langle X_\alpha : \alpha < \omega _1\rangle$ is a collection of subsets of $[\omega _1]^{\omega _1}$ with the property that $\bigcup _{\alpha < \omega _1}X_\alpha = [\omega _1]^{\omega _1}$, then there is an $\alpha < \omega _1$ so that $X_\alpha$ and $[\omega _1]^{\omega _1}$ are in bijection.

We show that under $\mathsf {AD}_\mathbb {R}$ everywhere $[\omega _1]^{<\omega _1}$-club uniformization holds which is the following statement: Let $\mathsf {club}_{\omega _1}$ denote the collection of club subsets of $\omega _1$. Suppose $R \subseteq [\omega _1]^{<\omega _1} \times \mathsf {club}_{\omega _1}$ is $\subseteq$-downward closed in the sense that for all $\sigma \in [\omega _1]^{<\omega _1}$, for all clubs $C \subseteq D$, $R(\sigma ,D)$ implies $R(\sigma ,C)$. Then there is a function $F \colon {\mathrm {dom}}(R) \rightarrow \mathsf {club}_{\omega _1}$ so that for all $\sigma \in {\mathrm {dom}}(R)$, $R(\sigma ,F(\sigma ))$.

We show that under $\mathsf {AD}$ almost everywhere $[{\omega _{1}}]^{<{\omega _{1}}}$-club uniformization holds which is the statement that for every $R \subseteq [{\omega _{1}}]^{<{\omega _{1}}} \times \mathsf {club}_{\omega _{1}}$ which is $\subseteq$-downward closed, there is a club $C$ and a function $F \colon {\mathrm {dom}}(R) \cap [C]^{<{\omega _{1}}}_* \rightarrow \mathrm {club}_{\omega _{1}}$ so that for all $\sigma \in {\mathrm {dom}}(R) \cap [C]^{<{\omega _{1}}}_*$, $R(\sigma ,F(\sigma ))$.