On the size of the minimum critical set of a Latin square

Abstract

A critical set in an $n \times n$ array is a set $C$ of given entries, such that there exists a unique extension of $C$ to an $n\times n$ Latin square and no proper subset of $C$ has this property. For a Latin square $L$, $\scs{L}$ denotes the size of the smallest critical set of $L$, and $\scs{n}$ is the minimum of $\scs{L}$ over all Latin squares $L$ of order $n$. We find an upper bound for the number of partial Latin squares of size $k$ and prove that $$n^2-(e+o(1))n^{10/6} \le \max \scs{L} \le n^2-\frac{\sqrt{\pi}}{2}n^{9/6}.$$ % This improves a result of N. Cavenagh (Ph.D. thesis, The University of Queensland, 2003) and disproves one of his conjectures. Also it improves the previously known lower bound for the size of the largest critical set of any Latin square of order $n$.