Matching perfetti e decomposizioni paradossali con la proprietà di Baire

We first introduce the definition of perfect matching, prove some elementary results on Borel graphs, and a strengthening of Hall’s theorem. We then introduce the concepts of paradoxical decomposition and equidecomposability, showing the ties between the two. After proving a few more technical results, we move to the main result of the thesis, which is taken from "A. Marks, S. Unger, Baire measurable paradoxical decompositions via matchings, Advances in Mathematics, 2015": if a group acts on a Polish space by Borel automorphisms and it has a paradoxical decomposition, then it also has one whose pieces have the property of Baire. Finally, we highlight the differences between the concepts of paradoxical decomposition and equidecomposability by showing that the corresponding result on the existence of equidecompositions with pieces having the property of Baire might fail. ​