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Register a new userOn the distances between Latin squares and the smallest defining set size
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In this note we show that for each Latin square $L$ of order $ngeq 2$, there exists a Latin square $L eq L$ of order $n$ such that $L$ and $L$ differ in at most $8sqrt{n}$ cells. Equivalently, each Latin square of order $n$ contains a Latin trade of size at most $8sqrt{n}$. We also show that the size of the smallest defining set in a Latin square is $Omega(n^{3/2})$. %That is, there are constants $c$ and $n_0$ such that for any $n>n_0$ the size of the smallest defining %set of order $n$ is at least $cn^{3/2}$.
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We prove a conjecture by Garbe et al. [arXiv:2010.07854] by showing that a Latin square is quasirandom if and only if the density of every 2x3 pattern is 1/720+o(1). This result is the best possible in the sense that 2x3 cannot be replaced with 2x2 or 1xN for any N.
We develop a limit theory of Latin squares, paralleling the recent limit theories of dense graphs and permutations. We introduce a notion of density, an appropriate version of the cut distance, and a space of limit objects - so-called Latinons. Key results of our theory are the compactness of the limit space and the equivalence of the topologies induced by the cut distance and the left-convergence. Last, using Keevashs recent results on combinatorial designs, we prove that each Latinon can be approximated by a finite Latin square.
We introduce a notion of parity for transversals, and use it to show that in Latin squares of order $2 bmod 4$, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving $E_1,dots, E_n$, where $E_i$ is the number of diagonals of a given Latin square that contain exactly $i$ different symbols. Let $A(imid j)$ denote the matrix obtained by deleting row $i$ and column $j$ from a parent matrix $A$. Define $t_{ij}$ to be the number of transversals in $L(imid j)$, for some fixed Latin square $L$. We show that $t_{ab}equiv t_{cd}bmod2$ for all $a,b,c,d$ and $L$. Also, if $L$ has odd order then the number of transversals of $L$ equals $t_{ab}$ mod 2. We conjecture that $t_{ac} + t_{bc} + t_{ad} + t_{bd} equiv 0 bmod 4$ for all $a,b,c,d$. In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a $k$-regular bipartite graph on $2n$ vertices is divisible by $4$ when $n$ is odd and $kequiv0bmod 4$. We also show that $${rm per}, A(a mid c)+{rm per}, A(b mid c)+{rm per}, A(a mid d)+{rm per}, A(b mid d) equiv 0 bmod 4$$ for all $a,b,c,d$, when $A$ is an integer matrix of odd order with all row and columns sums equal to $kequiv2bmod4$.
In this note, we study large deviations of the number $mathbf{N}$ of intercalates ($2times2$ combinatorial subsquares which are themselves Latin squares) in a random $ntimes n$ Latin square. In particular, for constant $delta>0$ we prove that $Pr(mathbf{N}le(1-delta)n^{2}/4)leexp(-Omega(n^{2}))$ and $Pr(mathbf{N}ge(1+delta)n^{2}/4)leexp(-Omega(n^{4/3}(log n)^{2/3}))$, both of which are sharp up to logarithmic factors in their exponents. As a consequence, we deduce that a typical order-$n$ Latin square has $(1+o(1))n^{2}/4$ intercalates, matching a lower bound due to Kwan and Sudakov and resolving an old conjecture of McKay and Wanless.
We consider the number of distinct distances between two finite sets of points in ${bf R}^k$, for any constant dimension $kge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O(1)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then $$ Omegaleft(minleft{ n^{2/3}m^{2/3},; frac{n^{10/11}m^{4/11}}{log^{2/11}m},; n^2,; m^2right}right) . $$ Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them.
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