2013/05/24 by Dilip Raghavan, Raghavan, Dilip, Stevo Todorcevic +1 · 2 citations
Mathematics · #FOS: Mathematics #Logic (math.LO) #math.LO
paper · pdf · doi:10.48550/arxiv.1305.5783
submitted
arxiv created 2013/05/24 · arxiv updated 2013/05/27
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 \mathfrakx such that the statement that \mathfrakx > 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 ℵ1. 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 \mathbbS. 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 \mathbbS. As a consequence we conclude that after forcing with the coherent Suslin tree \mathbbS 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 \mathbbS cannot be improved by showing in ZFC that \mathbbS \not→ (ℵ1, ω+2)2.