On the Ryser-Buraldi-Stein conjecture

A Latin square of order n is an n by n grid filled with n different symbols so that every symbol occurs exactly once in each row and each column, while a transversal in a Latin square is a collection of cells which share no row, column or symbol. The Ryser-Brualdi-Stein conjecture states that every Latin square of order n should have a transversal with n-1 elements, and one with n elements if n is odd. In 2020, Keevash, Pokrovskiy, Sudakov and Yepremyan improved the long-standing best known bounds on this conjecture by showing that a transversal with n-O(log n/loglog n) elements exists in any Latin square of order n. In this talk, I will discuss how to show, for large n, that a transversal with n-1 elements always exists.