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- W2963025237 abstract "Assuming the P-ideal dichotomy, we attempt to isolate those cardinal char- acteristics 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 >ω 1 is equivalent to the statement that 1, ω, ω1, ω × ω1 ,a nd ( ω1) <ω are the only cofinal types of directed sets of size at most ℵ1. We investigate the corresponding prob- lem for the partition relation ω1 → (ω1 ,α ) 2 for all α<ω 1. To this effect, we investigate partition relations for pairs of comparable elements of a coherent Suslin tree S.W e 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<ω 1 .W e prove that this positive partition relation for S cannot be improved by showing in ZFC that S �→ (ℵ1 ,ω +2 ) 2 ." @default.
- W2963025237 created "2019-07-30" @default.
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- W2963025237 date "2014-01-01" @default.
- W2963025237 modified "2023-09-29" @default.
- W2963025237 title "Combinatorial dichotomies and cardinal invariants" @default.
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- W2963025237 doi "https://doi.org/10.4310/mrl.2014.v21.n2.a13" @default.
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