Almost all Steiner triple systems have perfect matchings [article]

Matthew Kwan
2020 arXiv   pre-print
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
more » ... 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.
arXiv:1611.02246v5 fatcat:lsge56xdczgm5mzmd4u57dkare