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

TL;DR

This paper disproves Stein's Equi-n-square Conjecture by constructing arrays that lack large partial transversals, showing the conjecture does not hold universally.

Contribution

It provides a counterexample to Stein's conjecture, demonstrating that the conjecture is false for sufficiently large arrays.

Findings

Counterexamples exist for large arrays without partial transversals of size n - (1/42) ln n

Stein's Equi-n-square Conjecture is false in general

Constructs arrays with no large partial transversals

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 $n-\frac{1}{42}\ln n$.