Borel sets with large squares by

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^א0-square and even a perfect square, and also for ⊗′_μ if ψ ∈ L_ω1,ω has a model of cardinality μ then it has a model of cardinality continuum generated in a "nice", "absolute" way. Assuming MA+2^א0 > μ for transparency, those three conditions (⊕_μ, ⊗_μ and ⊗′_μ) are equivalent, and from this we deduce that e.g. ∧_α<ω1 [2^א0 ≥ א_α ⇒ ¬⊗_אα], and also that min{μ : ⊗_μ}, if < 2^א0, has cofinality א₁. We also deal with Borel rectangles and related model-theoretic problem.