Non-Glimm–Effros equivalence relations at second projective level

Vladimir Kanovei

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Mutually generic classes and incompatible expansions

Matt Kaufmann (1984)

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Borel sets with large squares

Saharon Shelah (1999)

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For a cardinal μ we give a sufficient condition $\oplus_{μ}$ (involving ranks measuring existence of independent sets) for: $\otimes_{μ}$ if a Borel set B ⊆ ℝ × ℝ contains a μ-square (i.e. a set of the form A × A with |A| =μ) then it contains a $2^{ℵ_{0}}$ -square and even a perfect square, and also for $\otimes_{μ}^{'}$ if $ψ \in L_{ω_{1}, ω}$ has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way. Assuming $M A + 2^{ℵ_{0}} > μ$ for transparency, those three conditions ( $\oplus_{μ}$ , $\otimes_{μ}$ and $\otimes_{μ}^{'}$ ) are equivalent, and from this we...