Descriptive complexity of countable unions of Borel rectangles

Abstract

We give, for each countable ordinal $\xi \geq 1$, an example of a ${\bf\Delta}^0_2$ countable union of Borel rectangles that cannot be decomposed into countably many ${\bf\Pi}^0_\xi$ rectangles. In fact, we provide a graph of a partial injection with disjoint domain and range, which is a difference of two closed sets, and which has no ${\bf\Delta}^0_\xi$-measurable countable coloring.