Transversals in Row-Latin Rectangles

It is shown that anm×nrow-latin rectangle with symbols in {1, 2, …, k},k⩾n, has a transversal wheneverm⩾2n−1, and that this lower bound formis sharp. Several applications are given. One is the construction of mappings which are generalizations of complete mappings. Another is the proof of a conjecture of Dillon on the existence of difference sets in groups of order 22s+2with elementary abelian normal subgroups of order 2s+1.