Substructures in Latin squares

Abstract

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 × 2 Latin subsquare) is at least ${({e^{- 9/4}}n - o(n))^{{n^2}}}$. Equivalently, $\Pr [{\bf{N}} = 0] \ge {e^{- {n^2}/4 - o({n^2})}} = {e^{- (1 + o(1)){\mathbb{E}\bf{N}}}}$, where 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 < δ ≤ 1 we have $\Pr [{\bf{N}} \le (1 - \delta){\mathbb{E}\bf{N}}] = \exp (- \Theta ({n^2}))$ and for any constant δ > 0 we have $\Pr [{\bf{N}} \ge (1 + \delta){\mathbb{E}\bf{N}}] = \exp (- \Theta ({n^{4/3}}log\,n))$.

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 × 2 submatrices with the same arrangement of symbols) is of order n⁴, 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.