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 \(n\times n\) array is a selection of \(n\) different symbols from different rows and different columns. We prove that every \(n \times n\) Latin array containing at least \((2-\sqrt{2}) n^2\) distinct symbols has a transversal. Also, every \(n \times n\) row-Latin array containing at least \(\frac14(5-\sqrt{5})n^2\) 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.