Combinatorial dichotomies and cardinal invariants

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 $\aleph$ such that the statement that $\aleph \gt \omega_1$ is equivalent to the statement that $1, \omega, \omega_1, \omega \times \omega_1$, and $[ \omega_1 ]^{\lt \omega}$ 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 $\alpha \lt \omega_1$. 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 $\alpha \lt \omega_1$. We prove that this positive partition relation for $\mathbb{S}$ cannot be improved by showing in ZFC that $\mathbb{S} \not \to (\aleph_1, \omega + 2)^2$.

Keywords

combinatorial dichotomies, partition relation, P-ideal dichotomy, cardinal invariants, coherent Suslin tree, Laver property

2010 Mathematics Subject Classification

03E05, 03E17, 03E65

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Published 1 August 2014