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 nn has a hole of order mm, then it is an easy observation that n≥2mn≥2m. More generally, if a set of tt incomplete mutually orthogonal latin squares of order nn have a common hole of order mm, then n≥(t+1)mn≥(t+1)m. In this article, we prove such sets of incomplete squares exist for all n,m≫0n,m≫0 satisfying n≥8(t+1)2mn≥8(t+1)2m.