Perfect matchings in -partite -graphs

Let H be an r -partite r -graph, all of whose sides have the same size n . Suppose that there exist two sides of H , each satisfying the following condition: the degree of each legal r − 1 -tuple contained in the complement of this side is strictly larger than n 2 . We prove that under this condition H must have a perfect matching. This answers a question of Kühn and Osthus.