Borel sets with large squares by

Abstract

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 2א0-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+2א0 > μ for transparency, those three conditions (⊕μ, ⊗μ and ⊗′μ) are equivalent, and from this we deduce that e.g. ∧α < 2א0, has cofinality א1. We also deal with Borel rectangles and related model-theoretic problem.