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 14(5-5)n2 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.