On the number of symbols that forces a transversal

TL;DR

This paper proves that large Latin arrays with at least half the maximum number of symbols contain a transversal, confirming a conjecture for large arrays and providing a specific upper bound on the number of symbols needed.

Contribution

It confirms Akbari and Alipour's conjecture for large arrays and establishes that $n^{399/200}$ symbols suffice to guarantee a transversal.

Findings

Confirmed the conjecture for large $n$

Established $n^{399/200}$ symbols as sufficient

Advanced understanding of transversals in Latin arrays

Abstract

Akbari and Alipour conjectured that any Latin array of order $n$ with at least $n^2/2$ symbols contains a transversal. We confirm this conjecture for large $n$, and moreover, we show that $n^{399/200}$ symbols suffice.