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arXiv · 2209.02180

Transversals in quasirandom latin squares

Sean Eberhard·Freddie Manners·Rudi Mrazović
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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 $\bigl((1 + o(1)) n / e^2\bigr)^n$ 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.

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Sean Eberhard, Freddie Manners, Rudi Mrazović. 2023-04-21. Transversals in quasirandom latin squares. https://doi.org/10.1112/plms.12538

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