Combinatorial dichotomies and cardinal invariants

Abstract

Assuming the P-ideal dichotomy, we attempt to isolate those cardinal characteristics of the continuum that are correlated with two well-known consequences of the proper forcing axiom. We find a cardinal invariant $\mathfrak{x}$ such that the statement that $\mathfrak{x} > {\omega}_{1}$ is equivalent to the statement that 1, $\omega$, ${\omega}_{1}$, $\omega \times {\omega}_{1}$, and ${\left[{\omega}_{1}\right]}^{< \omega}$ are the only cofinal types of directed sets of size at most ${\aleph}_{1}$. We investigate the corresponding problem for the partition relation ${\omega}_{1} \rightarrow ({\omega}_{1}, \alpha)^2$ for all $\alpha < {\omega}_{1}$. To this effect, we investigate partition relations for pairs of comparable elements of a coherent Suslin tree $\mathbb{S}$. We show that a positive partition relation for such pairs follows from the maximal amount of the proper forcing axiom compatible with the existence of $\mathbb{S}$. As a consequence we conclude that after forcing with the coherent Suslin tree $\mathbb{S}$ over a ground model satisfying this relativization of the proper forcing axiom, ${\omega}_{1} ~\rightarrow {({\omega}_{1}, \alpha)}^{2}$ for all $\alpha < {\omega}_{1}$. We prove that this positive partition relation for $\mathbb{S}$ cannot be improved by showing in $\mathrm{ZFC}$ that $\mathbb{S} \not\rightarrow ({\aleph}_{1}, \omega+2)^2$.