Exactness of the general upper bound for smallest critical sets

Determine whether the upper bound \(\mathrm{scs}(n)\leq \lfloor n^2/4\rfloor\) for the smallest size of a critical set among Latin squares of order \(n\) is always attained, equivalently, establish whether \(\mathrm{scs}(n)=\lfloor n^2/4\rfloor\) for every positive integer \(n\).

Background

For a Latin square $L$ of order $n$, $\mathrm{scs}(L)$ denotes the size of a smallest critical set of $L$, while $\mathrm{scs}(n)$ denotes the smallest such size over all Latin squares of order $n$. The paper recalls the general bound $\mathrm{scs}(n)\leq \lfloor n^2/4\rfloor$, together with known values for orders up to eight. The bound is known to be attained for some orders, and partial exact results are available for particular families such as back-circulant and symmetric Latin squares. The unresolved issue is whether attainment holds universally for every order.