Latin squares have been studied in Combinatorics since the early work of Euler in the 1780's. More recently, much progress has been made from the perspective of graph theory, allowing modern graph theory tools to be applied to an equivalent formulation using edge-coloured graphs. I will discuss in particular the study of transversals in Latin squares via this route, and the main extremal and probabilistic questions in the area:

i) How large a partial transversal can we find in any Latin square?

ii) How many disjoint transversals are we likely to find in a random Latin square?

More specifically: a Latin square of order n is an n by n grid filled with n symbols so that every symbol appears exactly once in each row and each column. Multiplication tables of finite groups are important examples, while they have links to 2-dimensional permutations, finite projective planes and error correcting codes. A partial transversal of a Latin square of order n is a collection of cells in the grid which share no row, column or symbol, and a transversal is a partial transversal with n cells.

Not every Latin square has a transversal. However, the Ryser-Brualdi-Stein conjecture originating from the 1960's says that every Latin square of order n should have a transversal if n is odd, and, if n is even, come only one cell short of having a transversal. I will discuss my recent work essentially resolving this by showing that, for sufficiently large n, any Latin square of order n has a partial transversal with n-1 cells.

We should expect to be able to find much more in a `typical' Latin square. In 1782, Euler conjectured that if n = 2 mod 4 then there are no Latin squares of order n which can be decomposed into n disjoint transversals, a conjecture disproved in the 1950's. Very recently, Candida Bowtell and I have shown that, moreover, a random Latin square of order n is very likely to be decomposable into transversals, in stark contrast to Euler's conjecture.