  For $\mu$ given latin squares of order $n$, they have {\sf $k$ intersection}
when they have $k$ identical cells and $n^2-k$ cells with mutually different
entries. For each $n\geq 1$ the set of integers $k$ such that there exist $\mu$
latin squares of order $n$ with $k$ intersection is denoted by $I^{\mu}[n]$. In
a paper by P. Adams et al. (2002), $I^3[n]$ is determined completely. In this
paper we completely determine $I^4[n]$ for $n\geq 16$. For $n \le 16$, we find
out most of the elements of $I^4[n]$.
