Combinatorial analysis (matrix problems, order theory)

Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

“…If we denote the minimum volume by m v (n), then the PLS(4) in Example 2.2 points to the conclusion that m v (n) n. However proving that the minimum m v (n) is n is non-trivial, with a number of papers appearing on this topic. Nosov, Sachkov and Tarakanov provide a brief review of these articles in [43], see also [2]. In 1970 Lindner, [37], solved the problem when the filled cells occur in less than n/2 rows and in 1981 Smetaniuk, see [48,13], gave a construction for the case where the filled cells intersect more than n/2 rows.…”