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Published online by Cambridge University Press: 20 November 2018
In the mid 1970's, Shelah formulated a weak version of ◊. This axiom Φ is a prediction principle for colorings of the binary tree of height ω1. Shelah and Devlin showed that Φ is equivalent to 2ℵ0 < 2ℵ1.
In this paper, we formulate Φp, a "Φ for partial colorings", show that both ◊* and Fleissner's “◊ for stationary systems” imply Φp, that ◊ does not imply Φp and that Φp does not imply CH.
We show that Φp implies that, in a normal first countable space, a discrete family of points of cardinality ℵ1 is separated.
This work has been supported by the Natural Sciences and Engineering Research Council of Canada.