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 x such that the statement that x < ω1 is equivalent to the statement that 1, ω, ω1, ω × ω1, and [ω1]<ω are the only cofinal types of directed sets of size at most N1. We investigate the corresponding problem for the partition relation ω1 → (ω1, α)2 for all α < ω1. 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, ω1 → (ω1, α)2 for all α < ω1. We prove that this positive partition relation for S cannot be improved by showing in ZFC that S → (N1, ω + 2)2.