Intercalates and Discrepancy in Random Latin Squares

An intercalate in a Latin square is a $2\times2$ Latin subsquare. Let $N$ be the number of intercalates in a uniformly random $n\times n$ Latin square. We prove that asymptotically almost surely $N\ge\left(1-o\left(1\right)\right)\,n^{2}/4$, and that $\mathbb{E}N\le\left(1+o\left(1\right)\right)\,n^{2}/2$ (therefore asymptotically almost surely $N\le fn^{2}$ for any $f\to\infty$). This significantly improves the previous best lower and upper bounds. We also give an upper tail bound for the number of intercalates in two fixed rows of a random Latin square. In addition, we discuss a problem of Linial and Luria on low-discrepancy Latin squares

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, $\Pr\left[\mathbf{N}=0\right]\ge e^{-n^{2}/4- (n^{2})}=e^{-(1+o(1))\mathbb{E}\mathbf{N}}$, 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 $\Pr[\mathbf{N}\le(1-\delta)\mathbb{E}\mathbf{N}]=\exp(-\Theta(n^{2}))$ and for any constant $\delta>0$ we have $\Pr[\mathbf{N}\ge(1+\delta)\mathbb{E}\mathbf{N}]=\exp(-\Theta(n^{4/3}(\log n)^{2/3}))$. Finally, we show that in almost all order-$n$ Latin squares, the number of cuboctahedra (i.e., the number of pairs of possibly degenerate $2\times2$ subsquares 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.Comment: 32 pages, 1 figur

Bounds on the number of small Latin subsquares

Let ζ(n, m) be the largest number of order m subsquares achieved by any Latin square of order n. We show that ζ(n, m) = Θ(n3 ) if m ∈ {2, 3, 5} and ζ(n, m) = Θ(n4 ) if m ∈ {4, 6, 9, 10}. In particular, 1 8 n3 + O(n2 ) . ζ(n, 2) . 1 4 n3 + O(n2 ) and 1 27 n3 + O(n5/2 ) . ζ(n, 3) . 1 18 n3 + O(n2 ) for all n. We find an explicit bound on ζ(n, 2d ) of the form Θ(nd+2 ) and which is achieved only by the elementary abelian 2-groups. For a fixed Latin square L let ζ∗ (n, L) be the largest number of subsquares isotopic to L achieved by any Latin square of order n. When L is a cyclic Latin square we show that ζ∗ (n, L) = Θ(n3 ). For a large class of Latin squares L we show that ζ∗ (n, L) = O(n3 ). For any Latin square L we give an in the interval (0, 1) such that ζ∗ (n, L) . Ω(n2+ ). We believe that this bound is achieved for certain squares L.</p