Fresh from the ArXiv
A proof of the Ryser-Brualdi-Stein conjecture for large even n by Richard Montgomery
A Latin square of order n is an n by n grid filled using n symbols so that each symbol appears exactly once in each row and column. A transversal in a 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 every Latin square of order n contains a transversal with n−1 cells, and a transversal with n cells if n is odd. Keevash, Pokrovskiy, Sudakov and Yepremyan recently improved the long-standing best known bounds towards this conjecture by showing that every Latin square of order n has a transversal with n−O(logn/loglogn) 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, every Steiner triple system of order n has a matching containing at least (n−4)/3 edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov and Yepremyan, who found such matchings with n/3−O(logn/loglogn) edges, and proves a conjecture of Brouwer from 1981 for large n.
Richard announced the result some months ago in a conference and yesterday the 71-page paper with the details of the proof was uploaded to the arXiv. Congratulations, Richard!
From right to left: Sherman Stein, Richard Brualdi, Marshal Hall and Herbert Ryser, and Richard Montgomery
I am thankful to Ryan Alweiss for telling me about the paper.