Subsquares in random Latin rectangles

TL;DR

This paper proves that large random Latin rectangles almost surely lack proper Latin subsquares of order 4 or more, and provides bounds on the expected number of smaller subsquares, advancing understanding of their structure.

Contribution

It establishes probabilistic bounds on the presence of Latin subsquares in large random Latin rectangles, confirming a conjecture and quantifying expected counts of small subsquares.

Findings

No proper Latin subsquares of order ≥4 with high probability

Expected number of order-3 subsquares is bounded

Expected number of order-2 subsquares is approximately half of the pairs

Abstract

Suppose that $k$ is a function of $n$ and $n\to\infty$. We show that with probability $1-O(1/n)$, a uniformly random $k\times n$ Latin rectangle contains no proper Latin subsquare of order $4$ or more, proving a conjecture of Divoux, Kelly, Kennedy and Sidhu. We also show that the expected number of subsquares of order 3 is bounded and find that the expected number of subsquares of order 2 is $\binom{k}{2}(1/2+o(1))$ for all $k\le n$.