Counting Near-Spanning Matchings in Latin Squares and Steiner Triple Systems

Montgomery recently proved that for sufficiently large $n$, every Latin square of order $n$ has a partial transversal with $n-1$ cells, and every Steiner triple system of order $n$ has a matching with $\lfloor n/3\rfloor-1$ edges, thus confirming the Ryser--Brualdi--Stein conjecture for even $n$ and