A new bound on the size of the largest critical set in a Latin square (Q1394806)

A partial Latin square of order \(n\) is called a critical set if it can be completed uniquely to a Latin square of the same order and none of its proper subsets has the same property. The authors obtain the new upper bound \(n^2-3n+3\) on the largest possible size of a critical set of order \(n\). Some small examples of large critical sets are given and some conjectures and problems are discussed as well.