Transversals in Latin arrays with many distinct symbols
An array is row-Latin if no symbol is repeated within any row.
An array is Latin if it and its transpose are both row-Latin. A
transversal in an n × n array is a selection of n
different symbols from different rows and different columns. We prove
that every n × n Latin array containing at least (2-√2)
n2 distinct symbols has a transversal. Also, every n × n
row-Latin array containing at least (5-√5)n2/4 distinct
symbols has a transversal. Finally, we show by computation that every
Latin array of order 7 has a transversal, and we describe all
smaller Latin arrays that have no transversal.