The four-way intersection problem for latin squares

For $μ$ 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 $μ$ latin squares of order $n$ with $k$ intersection is denoted by $I^μ[n]$. 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 $I^4[n]$.

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