The Intersection Problem for Semi-Symmetric Latin Squares

Abstract

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 $$\begin{array}{rl}J[3]& =\{0,9\},J[4]=\{0,1,3,4,9,12,16\},\\ J[5]& =\{0,1,3,4,6,7,9,10,12,13,15,18,21,25\},\\ J[6]& =\{0,1,2,\dots ,23,24,26,27,28,29,32,36\},\text{ and}\\ J[v]& =\{0,1,2,\dots ,v^2\}\setminus \{v^2-1,v^2-2,v^2-3,v^2-5,v^2-6\}\end{array}$$
for each $v\ge 7$.