Almost all Steiner triple systems have perfect matchings

Understand

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.