Transversals in quasirandom latin squares

Abstract

A transversal in an $n\times n$ latin square is a collection of $n$ entries not repeating any row, column, or symbol. Kwan showed that almost every $n\times n$ latin square has $((1+o(1))n/e^2{)}^n$ transversals as $n\to \infty$. Using a loose variant of the circle method we sharpen this to $(e^{-1/2}+o(1))n{!}^2/n^n$. Our method works for all latin squares satisfying a certain quasirandomness condition, which includes both random latin squares with high probability as well as multiplication tables of quasirandom groups.

26 pages, 3 figures. Final version incorporating referee's comments. To appear in Proceedings of the London Mathematical Society

References in corpus (1)

• An asymptotic for the Hall--Paige conjecture

Cited by in corpus (1)

• Every latin hypercube of order 5 has transversals