Intercalates and Discrepancy in Random Latin Squares

TL;DR

This paper investigates the number of intercalates in random Latin squares, establishing new bounds on their typical quantity and providing tail bounds, thereby advancing understanding of their structural properties.

Contribution

It provides improved asymptotic bounds on the number of intercalates in random Latin squares and addresses a problem on low-discrepancy Latin squares.

Findings

Asymptotically almost surely, the number of intercalates is at least (1-o(1))n^2/4.

Expected number of intercalates is at most (1+o(1))n^2/2.

Derived an upper tail bound for intercalates in two fixed rows.

Abstract

An intercalate in a Latin square is a $2\times2$ Latin subsquare. Let $N$ be the number of intercalates in a uniformly random $n\times n$ Latin square. We prove that asymptotically almost surely $N\ge\left(1-o\left(1\right)\right)\,n^{2}/4$, and that $\mathbb{E}N\le\left(1+o\left(1\right)\right)\,n^{2}/2$ (therefore asymptotically almost surely $N\le fn^{2}$ for any $f\to\infty$). This significantly improves the previous best lower and upper bounds. We also give an upper tail bound for the number of intercalates in two fixed rows of a random Latin square. In addition, we discuss a problem of Linial and Luria on low-discrepancy Latin squares.