Department of Mathematics2024-11-0920200012-365X10.1016/j.disc.2020.1118352-s2.0-85078899203https://hdl.handle.net/20.500.14288/3557In this paper it is shown that any partial Latin square of order n can be embedded in a Latin square of order at most 16n2 which has at least 2n mutually orthogonal mates. Further, for any t⩾2, it is shown that a pair of orthogonal partial Latin squares of order n can be embedded in a set of t mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to n. A consequence of the constructions is that, if N(n) denotes the size of the largest set of MOLS of order n, then N(n2)⩾N(n)+2. In particular, it follows that N(576)⩾9, improving the previously known lower bound N(576)⩾8.pdfMathematicsJournal Articlehttps://doi.org/10.1016/j.disc.2020.111835528203800014Q3NOIR02846