Applying Generic Coding with Help to Uniformizations

This is a follow up to a paper by the author where the disjointness relation for (the graphs of) definable functions from ${}^{\omega }\omega$ to ${^\omega \omega}$isanalyzed\mathrm{.}Inthatpaper,foreach$a \in {^\omega \omega}$ we defined a Baire class one function $f_a^{GC} : {^\omega \omega} \to {^\omega \omega}$whichencoded$a$inacertainsense\mathrm{.}Given$g : {^\omega \omega} \to {^\omega \omega}$,let$\Psi(g)$bethestatementthat$g$ is disjoint from at most countably many of the functions $f_a^{GC}$. We show the consistency strength of $(\mathrm{\forall }g)\mathrm{\Psi }(g)$ is at most one inaccessible cardinal. We show that $A{D}^{+}$ implies $(\mathrm{\forall }g)\mathrm{\Psi }(g)$. Finally, we show that assuming large cardinals, $(\mathrm{\forall }g)\mathrm{\Psi }(g)$ holds in models of the form $L(\mathbb{R}[\mathcal{U}]$ where $\mathcal{U}$ is a selective ultrafilter on $\omega$.