Mutually orthogonal latin squares with large holes

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 $n\ge 8(t+1{)}^{2}m$.