Mutually orthogonal latin squares with large holes

• DOI: 10.1016/j.jspi.2014.10.004
• Publication Date: 2014-10-27T14:23:26Z
• Authors: Peter J. Dukes; Christopher M. va...
• Keywords: 05B15, 05D05; FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO); 0101 mathematics; 01 natural sciences

Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once. This definition extends naturally to `incomplete' latin squares each having a hole on the same rows, columns, and symbols. If an incomplete latin square of order $n$ has a hole of order $m$, then it is an easy observation that $n \ge 2m$. More generally, if a set of $t$ incomplete mutually orthogonal latin squares of order $n$ have a common hole of order $m$, then $n \ge (t+1)m$. In this article, we prove such sets of incomplete squares exist for all $n,m \gg 0$ satisfying $n \ge 8(t+1)^2 m$.