Quasirandom Latin squares

TL;DR

This paper proves a conjecture that characterizes quasirandom Latin squares through the uniform density of 2x3 patterns, establishing a precise and optimal criterion for quasirandomness.

Contribution

It establishes a necessary and sufficient condition for Latin squares to be quasirandom based on 2x3 pattern densities, resolving a conjecture by Garbe et al.

Findings

Latin squares are quasirandom iff 2x3 pattern density is 1/720+o(1)

The 2x3 pattern density condition is optimal and cannot be replaced by smaller patterns

The result confirms the conjecture by Garbe et al.

Abstract

We prove a conjecture by Garbe et al. [arXiv:2010.07854] by showing that a Latin square is quasirandom if and only if the density of every 2x3 pattern is 1/720+o(1). This result is the best possible in the sense that 2x3 cannot be replaced with 2x2 or 1xN for any N.