Transversals in quasirandom latin squares

Abstract

A transversal in an n×n$n \times n$ latin square is a collection of n$n$ entries not repeating any row, column, or symbol. Kwan showed that almost every n×n$n \times n$ latin square has (1+o(1))n/e2n$\bigl ((1 + o(1)) n / e^2\bigr )^n$ transversals as n→∞$n \rightarrow \infty$ . Using a loose variant of the circle method we sharpen this to (e−1/2+o(1))n!2/nn$(e^{-1/2} + o(1)) n!^2 / n^n$ . 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.