Definable combinatorics at the first uncountable cardinal.

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 \vert[{\omega _{1}}]^{<{\omega _{1}}}\vert < \vert[{\omega _{1}}]^{\omega _{1}}\vert 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)). [ABSTRACT FROM AUTHOR]