Descriptive complexity of countable unions of Borel rectangles

We give, for each countable ordinal $ξ\geq 1$, an example of a ${\bfΔ}^0_2$ countable union of Borel rectangles that cannot be decomposed into countably many ${\bfΠ}^0_ξ$ 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 ${\bfΔ}^0_ξ$-measurable countable coloring.

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