The Intersection Problem for Semi-Symmetric Latin Squares

Let \(J[v]\) denote the set of numbers \(k\) such that there exist two semi-symmetric Latin squares (SSLS) of order \(v\) which have \(k\) entries in common. In this paper, we show that \begin{align*}
J[3] &= \{0, 9\}, J[4] = \{0, 1, 3, 4, 9, 12, 16\}, \\
J[5] &= \{0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 18, 21, 25\}, \\
J[6] &= \{0, 1, 2, \ldots, 23, 24, 26, 27, 28, 29, 32, 36\}, \text{ and} \\
J[v] &= \{0, 1, 2, \ldots, v^2\} \setminus \{v^2-1, v^2-2, v^2-3, v^2-5, v^2-6\}
\end{align*}
for each \(v \geq 7\).