On the Threshold Problem for Latin Boxes

TL;DR

This paper determines the probability threshold for the emergence of Latin boxes in random 0-1 arrays, extending known results for Latin squares and rectangles to more general cases.

Contribution

It provides an asymptotically tight threshold for Latin boxes in specific parameter regimes, generalizing the threshold for Latin squares and rectangles.

Findings

Threshold probability is  \, ext{log}(n)/n for certain Latin box configurations.

Results imply thresholds for Latin rectangles and edge-colorings of complete bipartite graphs.

Extends understanding of random combinatorial structures in high-dimensional arrays.

Abstract

Let $m \leq n \leq k$. An $m \times n \times k$ 0-1 array is a Latin box if it contains exactly $mn$ ones, and has at most one $1$ in each line. As a special case, Latin boxes in which $m = n = k$ are equivalent to Latin squares. Let $\mathcal{M}(m,n,k;p)$ be the distribution on $m \times n \times k$ 0-1 arrays where each entry is $1$ with probability $p$, independently of the other entries. The threshold question for Latin squares asks when $\mathcal{M}(n,n,n;p)$ contains a Latin square with high probability. More generally, when does $\mathcal{M}(m,n,k;p)$ support a Latin box with high probability? Let $\varepsilon>0$. We give an asymptotically tight answer to this question in the special cases where $n=k$ and $m \leq \left(1-\varepsilon \right) n$, and where $n=m$ and $k \geq \left(1+\varepsilon \right) n$. In both cases, the threshold probability is $\Theta \left( \log \left( n \right) / n \right)$. This implies threshold results for Latin rectangles and proper edge-colorings of $K_{n,n}$.