Latin Transversals of Rectangular Arrays By

Let m and n be integers, 2 ≤ m ≤ n. An m by n array consists of mn cells, arranged in m rows and n columns, and each cell contains exactly one symbol. A transversal of an array consists of m cells, one from each row and no two from the same column. A latin transversal is a transversal in which no symbol appears more than once. We investigte L(m, n), the largest integer such that if each symbol in an m by n array appears at most L(m, n) times, then the array must have a latin transversal. We will obtain upper and lower bounds on L(m, n) and also determine L(2, n) and L(3, n). The proof depends on a general construction due to E. In each case an attempt to construct a latin transversal might as well begin with a 1 in the top row. The 2 must then be selected from the 2's in the second row, and the 3 from the 3's in the third row. Such choices do not extend to a latin transversal.