We show that the exclusivity (E) principle singles out the set of quantum correlations associated to any exclusivity graph assuming the set of quantum correlations for the complementary graph. Moreover, we prove that, for self-complementary graphs, the E principle, by itself (i.e., without further assumptions), excludes any set of correlations strictly larger than the quantum set. Finally, we prove that, for vertex-transitive graphs, the E principle singles out the maximum value for the quantum correlations assuming only the quantum maximum for the complementary graph. This opens the door for testing the impossibility of higher-than-quantum correlations in experiments. Introduction.-One of the most seductive scientific challenges in recent times is deriving quantum theory (QT) from first principles. The starting point is assuming general probabilistic theories allowing for correlations that are more general than those that arise in QT, and the goal is to find principles that pick out QT from this landscape of possible theories.