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                * Princeton Discrete Math Seminar *
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Speaker: Richard Montgomery (U. Warwick)

Thursday 27th February, 3:00 in Fine Hall 224.

Title: Large transversals in equi-n-squares

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

More precisely, in 1975 Stein conjectured that any n by n square in 
which each cell has one of n symbols, so that each symbol is used exactly 
n times, contains a set of n-1 cells which share no row, column or symbol. 
That is, he conjectured that every equi-n-square must contain a partial
transversal with n-1 cells. Pokrovskiy and Sudakov disproved this 
conjecture in 2019. I will discuss new work showing that, however, an 
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, 
Zach Hunter and Teo Petrov.

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