Almost all Steiner triple systems have perfect matchings

TL;DR

This paper proves that almost all Steiner triple systems of order divisible by 3 contain a perfect matching, establishing a maximum bound and comparing random systems to a triangle removal process.

Contribution

It introduces a general theorem relating random Steiner triple systems to the triangle removal process, showing most systems have maximum perfect matchings.

Findings

Almost all Steiner triple systems have a perfect matching.

A general upper bound on the number of perfect matchings is established.

The methods can be adapted to other combinatorial designs like Latin squares.

Abstract

We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a Steiner triple system and show that almost all Steiner triple systems essentially attain this maximum. We accomplish this via a general theorem comparing a uniformly random Steiner triple system to the outcome of the triangle removal process, which we hope will be useful for other problems. Our methods can also be adapted to other types of designs; for example, we sketch a proof of the theorem that almost all Latin squares have transversals.