Borel sets with large squares
  This is a slightly corrected version of an old work.   For a cardinal $\mu$ we give a sufficient condition $\oplus_\mu$ (involving ranks measuring existence of independent sets) for:   $\otimes_\mu$ if a Borel set $B\subseteq \mathbb{R} \times \mathbb{R}$ contains a $\mu$-square (i.e. a set of the form $A \times A$, with $|A| =\mu)$ then it contains a $2^{\aleph_0}$-square and even a perfect square.   And also for   $\otimes'_\mu$ if $\psi\in L_{\omega_1, \omega}$ has a model of cardinality $\mu$ then it has a model of cardinality continuum generated in a ``nice", ``absolute" way.   Assuming $\mathrm{MA}+ 2^{\aleph_0}>\mu$ for transparency, those three conditions ($\oplus_\mu,\otimes_\mu$ and $\otimes'e_\mu$) are equivalent, and by this we get e.g. $\bigwedge\limits_{\alpha < \omega_1} [2^{\aleph_0} \ge \aleph_\alpha \Rightarrow \neg \otimes_{\aleph_\alpha}$], and also $\min\{\mu: \otimes_\mu\}$ has cofinality $\aleph_1$ if it is $<2^{\aleph_0}$.   We deal also with Borel rectangles and related model theoretic problems. 
Saharon Shelah