Universal probability bounds for partial Latin squares

Abstract

This paper studies the probability of substructures occurring in random Latin squares. Our main result states that if $\hat{I}\pm ,{\hat{I}}^2>0$ are such that $2\hat{I}\pm +{\hat{I}}^2<1$, then there are positive constants $\hat{I}´=\hat{I}´(\hat{I}\pm ,{\hat{I}}^2)$ and $\hat{I}”=\hat{I}”(\hat{I}\pm ,{\hat{I}}^2)$ such that if $P$ is a partial Latin square of order $n$ with $k=k(n)$ non-empty cells occupying at most $\hat{I}\pm n$ rows and ${\hat{I}}^{2}n$ columns, the probability that a random Latin square of order $n$ contains $P$ lies between $(\hat{I}´/n{)}^k$ and $(\hat{I}”/n{)}^k$. We apply this result to subsquares in random Latin squares to obtain the first proof of the fact that the expected number of subsquares of order $3$ in a random Latin square of order $n$ is non-vanishing as $n\to \infty$. We are also able to provide the best known asymptotics for the expected number of subsquares of order $a$ in a random Latin square of order $n$ when $2<a=o(n^{1/2})$. Finally, we discuss the implications of our result on other configurations in random Latin squares as well as on completions of partial Latin squares.

Minor typos corrected and one proof elaborated on