The graph G_0 was introduced by Kechris-Solecki-Todorcevic in the late 90s,
and has since turned into an essential object in descriptive set theory. In
joint work with Richard Ketchersid, we prove a version of the G_0-dichotomy
in models of AD^+. This is then used to establish that the quotient by the
equivalence relation E_0 is a successor of R, a result previously known
under AD_R, but (perhaps surprisingly) not in L(R).