Enumerating extensions of mutually orthogonal Latin squares

Two n×n Latin squares L1,L2 are said to be orthogonal if, for every ordered pair (x, y) of symbols, there are coordinates (i, j) such that L1(i,j)=x and L2(i,j)=y. A k-MOLS is a sequence of k pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed k, log-asymptotically tight bounds on the number of k-MOLS. To study the situation when k grows with n, we bound the number of ways a k-MOLS can be extended to a (k+1)-MOLS. These bounds are again tight for constant k, and allow us to deduce upper bounds on the total number of k-MOLS for all k. These bounds are close to tight even for k linear in n, and readily generalise to the broader class of gerechte designs, which include Sudoku squares

Embedding partial Latin squares in Latin squares with many mutually orthogonal mates

In this paper it is shown that any partial Latin square of order $n$ can be embedded in a Latin square of order at most $16n^2$ which has at least $2n$ mutually orthogonal mates. Further, for any $t\geq 2$, 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 $N(n^2)\geq N(n)+2$. In particular, it follows that $N(576)\ge 9$, improving the previously known lower bound $N(576)\ge 8$

On the number of transversals in latin squares

The logarithm of the maximum number of transversals over all latin squares of order $n$ is greater than $\frac{n}{6}(\ln n+ O(1))$