On the distances between Latin squares and the smallest defining set size

TL;DR

This paper demonstrates that Latin squares of order n are closely related through small trades and establishes a lower bound on the size of their smallest defining sets, advancing understanding of their combinatorial structure.

Contribution

It proves the existence of small Latin trades within any Latin square and establishes a lower bound of order n^{3/2} for the smallest defining set size.

Findings

Existence of Latin squares differing in at most 8√n cells

Small Latin trades of size at most 8√n exist in any Latin square

Smallest defining set size is at least proportional to n^{3/2}

Abstract

In this note we show that for each Latin square $L$ of order $n\geq 2$, there exists a Latin square $L'\neq L$ of order $n$ such that $L$ and $L'$ differ in at most $8\sqrt{n}$ cells. Equivalently, each Latin square of order $n$ contains a Latin trade of size at most $8\sqrt{n}$. 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 $cn^{3/2}$.