A NOTE ON THE CONSTRUCTION OF LARGE SET OF LATIN SQUARES WITH ONE ENTRY IN COMMON

A latin square of order $n$ is an $n \times n$ array such that each of the integers $1, 2, 3, \cdots, n$ occurs exactly once in each row and each column. A large set of latin squares of order $n$ having only one entry in common is a maximum set of latin squares of order $n$ such that each pair of them contains exactly one fixed entry in common. In this paper, we prove that a large set of latin squares of order $n$ having only one entry in common has $n - 1$ latin squares for each positive integer $n$, $n \ge 4$.