Uniformization of ladder system colorings and stationary precaliber forcings
Yushiro Aoki
TL;DR
The paper investigates how ladder-system colorings and their uniformization interact with fragments of forcing axioms, focusing on stationary precaliber $\aleph_1$ via ${\mathsf{stat}}{-}{\text{pc}}$ and its separation from $\sigma$-linked axioms. It develops stationary refinements of classic forcing properties, defines relevant posets ${\mathsf{stat}_E}$-pc and ${\mathsf{stat}}{-}{\text{(K)}}$, and proves preservation under iterations and connections to MA. Through the study of ladder-system colorings and Aronszajn trees, including SS/uniformizable notions and full-$T$-uniformization, the authors establish that ${\mathrm{MA}}_{\aleph_1}(\text{$\sigma$-linked})$ does not imply ${\mathrm{MA}}_{\aleph_1}({\mathsf{stat}}{-}{\text{pc}})$, while ${\mathrm{MA}}_{\aleph_1}({\mathsf{stat}}{-}{\text{pc}})$ enforces strong uniformization consequences on stationary parts and trees. The results delineate the strength of forcing axioms and their impact on uniformization phenomena, advancing the understanding of how combinatorial colorings interact with fragmented axioms. The work has implications for the structure of ladder colorings, Aronszajn trees, and related Whitehead-type problems.
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 $σ$-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 $σ$-linked posets.