Descriptive complexity of countable unions of Borel rectangles

We give, for each countable ordinal ξ⩾1, an example of a Δ20 countable union of Borel rectangles that cannot be decomposed into countably many Πξ0 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 Δξ0-measurable countable coloring.