Descriptive complexity of countable unions of Borel rectangles (Q2440858)

In this paper, the authors prove for each \(1\leq\xi<\omega_1\) the existence of a \(\Pi^0_2\)-set on \(\omega^\omega\times\omega^\omega\) which is not a countable union of rectangles of the form \(\Sigma^0_\xi\times\Delta^1_1\). The authors also present that for each \(1\leq\xi<\omega_1\) there exists a subset of \(\omega^\omega\times\omega^\omega\) which is the difference of two closed sets and has no \(\Delta^0_\xi\)-measurable countable coloring. To do so, the authors first generalize a technical theorem [\textit{T. Mátrai}, Fund. Math. 183, No. 2, 157--168 (2004; Zbl 1071.26004), Theorem 4] in \(\omega^\omega\) instead of \(2^\omega\).