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 $\mathbb{S} \not\rightarrow ({\aleph}{1}, \omega+2)^2$.