Borel Sets with Large Squares
نویسنده
چکیده
For a cardinal μ we give a sufficient condition ⊕μ (involving ranks measuring existence of independent sets) for: ⊗μ: if a Borel set B ⊆ R × R contains a μ-square (i.e. a set of the form A × A, |A| = μ) then it contains a 20 -square and even a perfect square. And also for ⊗′μ: if ψ ∈ Lω1,ω has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute”way. Assuming MA+20 > μ for transparency, those three conditions (⊕μ, ⊗μ and ⊗ ′ μ) are equivalent, and by this we get e.g. ∧
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