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A Latin square of order $n$ is an $n$ by $n$ grid filled using $n$ symbols sothat each symbol appears exactly once in each row and column. A transversal ina Latin square is a collection of cells which share no symbol, row or column.The Ryser-Brualdi-Stein conjecture, with origins from 1967, states that everyLatin square of order $n$ contains a transversal with $n-1$ cells, and atransversal with $n$ cells if $n$ is odd. Keevash, Pokrovskiy, Sudakov andYepremyan recently improved the long-standing best known bounds towards thisconjecture by showing that every Latin square of order $n$ has a transversalwith $n-O(\log n/\log\log n)$ cells. Here, we show, for sufficiently large $n$,that every Latin square of order $n$ has a transversal with $n-1$ cells.  We also apply our methods to show that, for sufficiently large $n$, everySteiner triple system of order $n$ has a matching containing at least $(n-4)/3$edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov andYepremyan, who found such matchings with $n/3-O(\log n/\log\log n)$ edges, andproves a conjecture of Brouwer from 1981 for large $n$.

A proof of the Ryser-Brualdi-Stein conjecture for large even $n$