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