A $k$-plex in a Latin square is a selection of entries which has
exactly $k$ representatives from each row, column and symbol. The
$1$-plexes are transversals and have been studied since
Euler. In E-JC Vol.9, I proved the first theorems for general
$k$-plexes, and conjectured that for all even orders $n>4$ there is
a Latin square that has $3$-plexes but no transversal.
Much more recently, in joint work with Nick Cavenagh published in E-JC
Vol.24, we proved this conjecture and that there are very many
Latin squares without transversals. I will discuss these two
papers and the intervening history.