  We show that every locally finite bipartite Borel graph satisfying a
strengthening of Hall's condition has a Borel perfect matching on some comeager
invariant Borel set. We apply this to show that if a group acting by Borel
automorphisms on a Polish space has a paradoxical decomposition, then it admits
a paradoxical decomposition using pieces having the Baire property. This
strengthens a theorem of Dougherty and Foreman who showed that there is a
paradoxical decomposition of the unit ball in $\mathbb{R}^3$ using Baire
measurable pieces. We also obtain a Baire category solution to the dynamical
von Neumann-Day problem: if $a$ is a nonamenable action of a group on a Polish
space $X$ by Borel automorphisms, then there is a free Baire measurable action
of $\mathbb{F}_2$ on $X$ which is Lipschitz with respect to $a$.
