Note on random Latin squares and the triangle removal process

TL;DR

This paper explores the properties of random Latin squares, establishing key lemmas and their connection to the triangle removal process, to support analysis of large deviations in Latin squares.

Contribution

It provides analogues of lemmas from Steiner triple systems for Latin squares and links Latin squares to the triangle removal process, advancing understanding of their probabilistic structure.

Findings

Established lemmas for random Latin squares

Connected Latin squares to the triangle removal process

Supported analysis of large deviations in Latin squares

Abstract

This is a companion note to the paper "Almost all Steiner triple systems have perfect matchings (arXiv:1611.02246). That paper contains several general lemmas about random Steiner triple systems; in this note we record analogues of these lemmas for random Latin squares, which in particular are necessary ingredients for our recent paper "Large deviations in random Latin squares" (arXiv:2106.11932). Most important is a relationship between uniformly random order-$n$ Latin squares and the triangle removal process on the complete tripartite graph $K_{n,n,n}$.