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Borel sets with large squares

Saharon Shelah (1999)

Fundamenta Mathematicae

 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 with |A| =μ) then it contains a 2 0 -square and even a perfect square, and also for μ ' if ψ 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 0 > μ for transparency, those three conditions ( μ , μ and μ ' ) are equivalent, and from this we deduce that...

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