Subsquares in random Latin squares and rectangles

TL;DR

This paper investigates the likelihood of finding sparse partial Latin rectangles within random Latin squares and establishes that large Latin subsquares of order exceeding a certain threshold are asymptotically almost surely absent.

Contribution

It provides probabilistic bounds on the existence of sparse partial Latin rectangles and determines the asymptotic maximum size of Latin subsquares in random Latin squares.

Findings

Probability of sparse partial Latin rectangles is approximately $(rac{1 ext{±} ext{ } ext{small}}{n})^ ext{number of entries}$.

Large Latin subsquares of order greater than $c\sqrt{n ext{log} n}$ are almost surely absent in random Latin squares.

The results hold for sufficiently large $n$ and small $eta$, with explicit bounds derived.

Abstract

A $k \times n$ partial Latin rectangle is \textit{$C$-sparse} if the number of nonempty entries in each row and column is at most $C$ and each symbol is used at most $C$ times. We prove that the probability a uniformly random $k \times n$ Latin rectangle, where $k < (1/2 - \alpha)n$, contains a $\beta n$-sparse partial Latin rectangle with $\ell$ nonempty entries is $(\frac{1 \pm \varepsilon}{n})^\ell$ for sufficiently large $n$ and sufficiently small $\beta$. Using this result, we prove that a uniformly random order-$n$ Latin square asymptotically almost surely has no Latin subsquare of order greater than $c\sqrt{n\log n}$ for an absolute constant $c$.