On the spectrum of critical sets in latin squares of order 2

Suppose that L is a latin square of order m and P ⊆ L is a partial latin square. If L is the only latin square of order m which contains P , and no proper subset of P has this property, then P is a critical set of L. The critical set spectrum problem is to determine, for a given m, the set of integers t for which there exists a latin square of order m with a critical set of size t. We outline a partial solution to the critical set spectrum problem for latin squares of order 2. The back circulant latin square of even order m has a well-known critical set of size m/4, and this is the smallest known critical set for a latin square of order m. The abelian 2-group of order 2 has a critical set of size 4 − 3, and this is the largest known critical set for a latin square of order 2. We construct a set of latin squares with associated critical sets which are intermediate between the back circulant latin square of order 2 and the abelian 2-group of order 2.