Borel sets with large squares

 For a cardinal μ we give a sufficient condition ${\displaystyle {\oplus }_{\mu }}$ (involving ranks measuring existence of independent sets) for: ${\displaystyle {\otimes }_{\mu }}$ 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 ${\displaystyle 2^{{\aleph }_0}}$-square and even a perfect square, and also for ${\displaystyle \otimes {\prime }_{\mu }}$ if ${\displaystyle \psi \in L_{{\omega }_1,\omega }}$ has a model of cardinality μ then it has a model of cardinality continuum generated in a "nice", "absolute" way. Assuming ${\displaystyle MA+2^{{\aleph }_0}>\mu }$ for transparency, those three conditions (${\displaystyle {\oplus }_{\mu }}$, ${\displaystyle {\otimes }_{\mu }}$ and ${\displaystyle \otimes {\prime }_{\mu }}$) are equivalent, and from this we deduce that e.g. ${\displaystyle {\wedge }_{\alpha <{\omega }_1}[2^{{\aleph }_0}\ge {\aleph }_{\alpha }\Rightarrow ￢{\otimes }_{{\aleph }_{\alpha }}]}$, and also that ${\displaystyle min\{\mu :{\otimes }_{\mu }\}}$, if ${\displaystyle <2^{{\aleph }_0}}$, has cofinality ${\displaystyle {\aleph }_1}$.   We also deal with Borel rectangles and related model-theoretic problems.