On $\hat{\mathbf{I}}{\pounds }_3^1$- and $\hat{I}{\pounds }_4^1$-uniformization

Abstract

Assuming the consistency of $\mathsf{ZFC}$, we construct a model of set theory in which the boldface $\hat{\mathbf{I}}{\pounds }_3^1$-uniformization property holds, yet the lightface $\hat{I}{\pounds }_4^1$-uniformization property fails, separating these two principles for the first time. We also indicate how to create a universe where $\hat{I}{\pounds }_3^1$-uniformization holds, but $\hat{I}{\pounds }_4^1$-uniformization fails using inner models with large cardinals.