Uniformization of ladder system colorings and stationary precaliber forcings

Abstract

We investigate the relationship between variants of the uniformization property for ladder system colorings and fragments of Martin's Axiom. The well-known forcing properties of having precaliber $\aleph_1$ and being $\sigma$-centered correspond to uncountable refinement and countable decomposition into centered subsets, respectively, and the associated forcing axioms have been widely studied. In this paper, we focus on a forcing axiom for the property corresponding to stationary refinement, namely the stationary precaliber $\aleph_1$ property. Analogously, we observe that ladder system coloring uniformization also admits both stationary refinement and countable decomposition variants. We discuss the interaction between these uniformization properties and various forcing axioms. Through this analysis, we obtain as a main result the separation between the forcing axioms for stationary precaliber $\aleph_1$ and for $\sigma$-linked posets.