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Displaying similar documents to “On the splitting number and Mazurkiewicz's theorem”

Borel sets with large squares

Saharon Shelah (1999)

Fundamenta Mathematicae

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 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 ℵ 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 ℵ 0 > μ for transparency, those three conditions ( ⊕ μ , ⊗ μ and ⊗ μ ' ) are equivalent, and from this we...

Non-Glimm–Effros equivalence relations at second projective level

Vladimir Kanovei (1997)

Fundamenta Mathematicae

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A model is presented in which the Σ 2 1 equivalence relation xCy iff L[x]=L[y] of equiconstructibility of reals does not admit a reasonable form of the Glimm-Effros theorem. The model is a kind of iterated Sacks generic extension of the constructible model, but with an “ill“founded “length” of the iteration. In another model of this type, we get an example of a Π 2 1 non-Glimm-Effros equivalence relation on reals. As a more elementary application of the technique of “ill“founded Sacks iterations,...