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05 June Friday

## Liana Yepremyan ( London School of Economics and Political Science, University of Illinois at Chicago) Ryser’s conjecture and more

05 June 2020, 11:00 - 05 June 2020, 12:00
A Latin square of order $n$ is an $n \times n$ array filled with $n$ symbols such that each symbol appears only once in every row or column and a transversal is a collection of cells which do not share the same row, column or symbol.  The study of Latin squares goes back more than 200 years to the work of Euler. One of the most famous open problems in this area is a conjecture of Ryser, Brualdi and Stein from 60s which says that every Latin square  of order $n\times n$ contains a transversal of order $n-1$. A closely related problem is 40 year old conjecture of Brouwer  that every Steiner triple system of order $n$ contains a matching of size $(n-4)/3$. The third problem we’d like to mention asks how many distinct symbols in Latin arrays suffice to guarantee a full transversal? In this talk we discuss a novel approach to attack  these problems.