  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 $n$.
