31/May/2013, 13:30–15:00

Fields institute,Room 210

Speaker: Dilip Raghavan

Title: 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 $\chi$ such that the statement that $\chi> {\omega}_{1}$ is equivalent to the statement that $1$, $\omega$, ${\omega}_{1}$, $\omega \times {\omega}_{1}$, and ${[{\omega}_{1}]}^{< \omega}$ 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 $S$.

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 $S$.

As a consequence we conclude that after forcing with the coherent Suslin tree $S$ 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 $S$ cannot be improved by showing in ZFC that $S\not\rightarrow ({\aleph}_{1}, \omega+2)^2$.