Ladder system uniformization on trees I & II

Abstract

Given a tree $T$ of height $\omega_1$, we say that a ladder system colouring $(f_\alpha)_{\alpha\in \lim\omega_1}$ has a $T$-uniformization if there is a function $\varphi$ defined on a subtree $S$ of $T$ so that for any $s\in S_\alpha$ of limit height and almost all $\xi\in {dom} (f_\alpha)$, $\varphi(s\upharpoonright \xi)=f_\alpha(\xi)$. In sharp contrast to the classical theory of uniformizations on $\omega_1$, J. Moore proved that CH is consistent with the statement that any ladder system colouring has a $T$-uniformization (for any Aronszajn tree $T$). Our goal is to present a fine analysis of ladder system uniformization on trees pointing out the analogies and differences between the classical and this new theory. We show that if $S$ is a Suslin tree then (i) CH implies that there is a ladder system colouring without $S$-uniformization; (ii) the restricted forcing axiom $MA(S)$ implies that any ladder system colouring has an $\omega_1$-uniformization. For an arbitrary Aronszajn tree $T$, we show how diamond-type assumptions affect the existence of ladder system colourings without a $T$-uniformization. Furthermore, it is consistent that for any Aronszajn tree $T$ and ladder system $\mathbf C$ there is a colouring of $\mathbf C$ without a $T$-uniformization; however, and quite surprisingly, $\diamondsuit^+$ implies that for any ladder system $\mathbf C$ there is an Aronszajn tree $T$ so that any monochromatic colouring of $\mathbf C$ has a $T$-uniformization. We also prove positive uniformization results in ZFC for some well-studied trees of size continuum, and finish with a list of open problems.