Baire paradoxical decompositions need at least 6 pieces
Résumé
We prove that in certain cases, paradoxical decompositions of compact metric spaces using sets (or even $[0,1]$-valued functions) with the property of Baire modulo meager sets need more pieces than paradoxical decompositions with unrestricted pieces. In particular, any Baire paradoxical decomposition of the sphere $S^2$ using isometries needs at least six pieces.
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