A note on the McKelvey uncovered set and Pareto optimality

We consider the notion of Pareto optimality under the assumption that only the pairwise majority relation is known and show that the set of necessarily Pareto optimal alternatives coincides with the McKelvey uncovered set. As a consequence, the McKelvey uncovered set constitutes the coarsest Pareto optimal majoritarian social choice function. Moreover, every majority relation is induced by a preference profile in which the uncovered alternatives precisely coincide with the Pareto optimal ones. We furthermore discuss the structure of the McKelvey covering relation and the McKelvey uncovered set.