Minimum number of full transversals and perfect matchings conditional on existence

Determine the minimum number of full transversals among Latin squares of order n that have at least one full transversal, and the minimum number of m-edge matchings among Steiner triple systems of order n that have at least one such matching, where m=floor(n/3).

Background

The main theorems determine, up to a sharp asymptotic factor, the number of largest matchings that are guaranteed to exist in every Latin square and Steiner triple system. Full transversals in Latin squares and m-edge matchings in Steiner triple systems, however, may be absent altogether. The authors therefore identify the unresolved extremal question of how few such structures can occur among instances where at least one exists.

References

The situation of counting $n$-cell transversals or $m$-edge matchings is less clear, since the number of them could be zero. It is natural to ask how large the numbers must be once they exist. More precisely, what are $\min\bigl{T_n(L):T_n(L)>0\bigr}$ and $\min\bigl{N_m(S):N_m(S)>0\bigr}$?

— Counting Near-Spanning Matchings in Latin Squares and Steiner Triple Systems  (2609.11006 - Tang et al., 10 Sep 2026) in Section 6, subsection “Counting matchings of other sizes”