  We prove several results about substructures in Latin squares. First, we
explain how to adapt our recent work on high-girth Steiner triple systems to
the setting of Latin squares, resolving a conjecture of Linial that there exist
Latin squares with arbitrarily high girth. As a consequence, we see that the
number of order-$n$ Latin squares with no intercalate (i.e., no $2\times2$
Latin subsquare) is at least $(e^{-9/4}n-o(n))^{n^{2}}$. Equivalently,
$\mathbb{P}\left[\mathbf{N}=0\right]\ge
e^{-n^{2}/4-o(n^{2})}=e^{-(1+o(1))\mathbb{E}\mathbf{N}}$, where $\mathbf{N}$ is
the number of intercalates in a uniformly random order-$n$ Latin square.
  In fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the
general large-deviation problem for intercalates in random Latin squares, up to
constant factors in the exponent: for any constant $0<\delta\le1$ we have
$\mathbb{P}[\mathbf{N}\le(1-\delta)\mathbb{E}\mathbf{N}]=\exp(-\Theta(n^{2}))$
and for any constant $\delta>0$ we have
$\mathbb{P}[\mathbf{N}\ge(1+\delta)\mathbb{E}\mathbf{N}]=\exp(-\Theta(n^{4/3}\log
n))$.
  Finally, as an application of some new general tools for studying
substructures in random Latin squares, we show that in almost all order-$n$
Latin squares, the number of cuboctahedra (i.e., the number of pairs of
possibly degenerate $2\times2$ submatrices with the same arrangement of
symbols) is of order $n^{4}$, which is the minimum possible. As observed by
Gowers and Long, this number can be interpreted as measuring ``how
associative'' the quasigroup associated with the Latin square is.
