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 ${\omega }_1\to ({\omega }_1,\alpha {)}^2$ 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, ${\omega }_1\to {({\omega }_1,\alpha )}^2$ for all $\alpha <{\omega }_1$. We prove that this positive partition relation for $\mathbb{S}$ cannot be improved by showing in $\mathrm{ZFC}$ that $\mathbb{S}\nrightarrow ({\aleph }_1,\omega +2{)}^2$.

arXiv commentsubmitted