Small Latin arrays have a near transversal
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 x 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.