Substructures in Latin squares

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.