Descriptive complexity of countable unions of Borel rectangles.

Abstract: We give, for each countable ordinal , an example of a countable union of Borel rectangles that cannot be decomposed into countably many 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 -measurable countable coloring. [Copyright &y& Elsevier]