Orthogonal Latin squares with subsquares (Q792328)

Let LS(v,n) denote a pair of orthogonal latin squares of side v which contain a pair of common orthogonal subsquares of order n. It is known that such a pair can exist only if \(v\geq 3n\). It is shown here that for every integer \(n\geq 304\), an LS(v,n) exists if \(v>3n+6.\) It is also shown that if n is odd and \(n\geq 304\), then an LS(v,n) exists for all \(v\geq 3n\). A similar result is obtained for several infinite families for v even. The result greatly improves an earlier result of \textit{D. J. Crampin} ad \textit{A. J. W. Hilton} [J. Comb. Theory, Ser. A 19, 84-94 (1975; Zbl 0325.05012)].