Small Latin arrays have a near transversal

Abstract

A Latin array is a matrix of symbols in which no symbol occurs more than once within a row or within a column. A diagonal of an n × n array is a selection of n cells taken from different rows and columns of the array. The weight of a diagonal is the number of different symbols on it. We show via computation that every Latin array of order n 11 has a diagonal of weight at least n − 1. A corollary is the existence of near transversals in Latin squares of these orders.More generally, for all k 20 we compute a lower bound on the order of any Latin array that does not have a diagonal of weight at least n − k.