The number of distinct symbols in sections of rectangular arrays

We investigate transversals of rectangular arrays. For positive integers m and n  , where 2⩽m⩽n2⩽m⩽n an m by n array consists of mn cells arranged in m rows and n   columns. Each cell contains one symbol. When m=nm=n we speak of an array of order n. A section in the array consists of m cells, one from each row and no two from the same column. A transversal is a section whose m symbols are distinct. A partial transversal is a subset of a transversal. We investigate the existence in an array of a section with many different symbols, in particular the existence of a transversal.