Paradoxical decompositions into pieces with the Baire Property

I will present the following theorem due to Andrew Marks and Spencer Unger: Suppose G is a group of Borel automorphisms of a Polish space X. Then if X has a G-paradoxical decomposition into arbitrary pieces, then it also has a G-paradoxical decomposition into  pieces with the Baire Property. It is a consequence of a version of the Hall's marriage theorem for bipartite Borel graphs. These results come from a recent paper by Marks and Unger (``Baire paradoxical decompositions via matching", 
http://arxiv.org/pdf/1501.01690.pdf).