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.

我们证明了关于拉丁方子结构的几个结果。首先，我们解释如何将我们最近关于高周长斯坦纳三重系统的工作应用于拉丁方格的设置，解决 Linial 的猜想，即存在任意高周长的拉丁方格。因此，我们看到没有插值的n阶拉丁方阵（即没有 2 × 2 拉丁子方阵）的数量至少为\({({e^{- 9/4}}n - o(n ))^{{n^2}}}\)。等价地，\(\Pr [{\bf{N}} = 0] \ge {e^{- {n^2}/4 - o({n^2})}} = {e^{- (1 + o(1)){\mathbb{E}\bf{N}}}}\)，其中N是均匀随机顺序n拉丁方中的插值数。

事实上，扩展 Kwan、Sah 和 Sawhney 最近的工作，我们解决了随机拉丁方中插入的一般大偏差问题，直到指数中的常数因子：对于任何常数 0 < δ ≤ 1，我们有\ ( \ Pr [{\bf{N}} \le (1 - \delta){\mathbb{E}\bf{N}}] = \exp (- \Theta ({n^2}))\) 且对于任意常数δ > 0 我们有\(\Pr [{\bf{N}} \ge (1 + \delta){\mathbb{E}\bf{N}}] = \exp (- \Theta ({n^ {4/3}}log\,n))\)。

最后，作为研究随机拉丁方子结构的一些新通用工具的应用，我们表明，在几乎所有n阶拉丁方中，立方八面体的数量（即，可能退化的 2 × 2 子矩阵对的数量相同的符号排列）的阶数为n ⁴，这是可能的最小值。正如高尔斯和朗所观察到的，这个数字可以解释为衡量与拉丁方相关的拟群的“结合程度”。