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≥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≥(t+1)m. In this article, we prove such sets of
incomplete squares exist for all n,m≫0 satisfying n≥8(t+1)2m