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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 decomposedinto countably many ${\bf\Pi}^0_\xi$ rectangles. In fact, we provide a graph ofa partial injection with disjoint domain and range, which is a difference oftwo closed sets, and which has no ${\bf\Delta}^0_\xi$-measurable countablecoloring.

Descriptive complexity of countable unions of Borel rectangles