A counterexample to Stein's Equi-n-square Conjecture

Abstract

In 1975 Stein conjectured that in every $n\times n$ array filled with the numbers $1, \dots, n$ with every number occuring exactly $n$ times, there is a partial transversal of size $n-1$. In this note we show that this conjecture is false by constructing such arrays without partial transverals of size $n-\frac{1}{42}\ln n$.