Transversals in Latin Squares

TL;DR

This survey reviews recent progress in the study of transversals in Latin squares, highlighting new results and ongoing challenges in combinatorial design theory over the past decade.

Contribution

It compiles and analyzes recent findings on transversals in Latin squares, emphasizing developments and open problems from the last ten years.

Findings

Significant progress in existence results for transversals.

Identification of key open problems and conjectures.

New bounds and constructions for Latin square transversals.

Abstract

A Latin square is an $n$ by $n$ grid filled with $n$ symbols so that each symbol appears exactly once in each row and each column. A transversal in a Latin square is a collection of cells which do not share any row, column, or symbol. This survey will focus on results from the last decade which have continued the long history of the study of transversals in Latin squares.