Product Constructions For Critical Sets In Latin Squares

Let \( T \) be a partial Latin square. If there exist two distinct Latin squares \( M \) and \( N \) of the same order such that \( M \cap N = T \), then \( M \setminus T \) is said to be a Latin trade. For a given Latin square \( M \), it is possible to identify a subset of entries, termed a critical set, which intersects all Latin trades in \( M \) and is minimal with respect to this property.

Stinson and van Rees have shown that under certain circumstances, critical sets in Latin squares \( M \) and \( N \) can be used to identify critical sets in the direct product \( M \times N \). This paper presents a refinement of Stinson and van Rees’ results and applies this theory to prove the existence of two new families of critical sets.