Borel sets with large squares

Abstract

This is a slightly corrected version of an old work. For a cardinal $μ$ we give a sufficient condition $\oplus_μ$ (involving ranks measuring existence of independent sets) for: $\otimes_μ$ if a Borel set $B\subseteq \mathbb{R} \times \mathbb{R}$ contains a $μ$-square (i.e. a set of the form $A \times A$, with $|A| =μ)$ then it contains a $2^{\aleph_0}$-square and even a perfect square. And also for $\otimes'_μ$ if $ψ\in L_{ω_1, ω}$ has a model of cardinality $μ$ then it has a model of cardinality continuum generated in a ``nice", ``absolute" way. Assuming $\mathrm{MA}+ 2^{\aleph_0}>μ$ for transparency, those three conditions ($\oplus_μ,\otimes_μ$ and $\otimes'e_μ$) are equivalent, and by this we get e.g. $\bigwedge\limits_{α< ω_1} [2^{\aleph_0} \ge \aleph_α\Rightarrow \neg \otimes_{\aleph_α}$], and also $\min\{μ: \otimes_μ\}$ has cofinality $\aleph_1$ if it is $<2^{\aleph_0}$. We deal also with Borel rectangles and related model theoretic problems.