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The study of transversals in Latin squares has a long history,Ìýwhere the main extremal question is the subject of the well-known Ryser-Brualdi-Stein conjecture. Intriguingly, for a long time, it wasÌý not known whether this conjecture should hold given much weakerÌý conditions than those which define Latin squares. In 2019, PokrovskiyÌý and Sudakov disproved the related conjecture of Stein, but in this talkÌý I will discuss the extent to which it is true and give new upper andÌý lower bounds on the relevant extremal problem.Ìý

More precisely, in 1975 Stein conjectured that any n by n square inÌý which each cell has one of n symbols, so that each symbol is usedÌýexactly n times, contains a set of n-1 cells which share no row, columnÌýor symbol. That is, he conjectured that every equi-n-square must containÌýa partial transversal with n-1 cells. Pokrovskiy and Sudakov disproved this conjecture in 2019. I will discuss new work showing that, however,Ìýan approximate version of Stein's conjecture is true, and give new bounds in both directions on how large a partial transversal can be found in any equi-n square. This is joint work with Debsoumya Chakraborti, Micha Christoph, ZachÌýHunter and Teo Petrov.