We construct a model of ZFC with a singular cardinal κ such that every subset of κ in L(Vκ+1) has both the κ-Perfect Set Property and the U⃗-Baire Property. This is a higher analogue of Solovay's result for L(R). We obtain this configuration starting with large-cardinal assumptions in the realm of supercompactness, thus improving former theorems by Cramer, Shi and Woodin.
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Dimonte et al. (2024) studied this question.
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