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