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 x such that the statement that x > omega(1) is equivalent to the statement that 1, omega, omega(1), omega x 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) -> (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) -> (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 negated right arrow (aleph(1), omega + 2)(2).