A counting proof for the smallest critical set in a Latin square

Abstract

A critical set in a Latin square is a set of entries that determines the square uniquely and is minimal with this property. We prove that every partial Latin square of order $n$ with fewer than $\lfloor n^2/4\rfloor$ filled cells has an even number of completions. Consequently every critical set in every Latin square of order $n$ has at least $\lfloor n^2/4\rfloor$ entries. Since the cyclic Latin square has a critical set of exactly this size, the minimum size of a critical set over all Latin squares of order $n$ is $\lfloor n^2/4\rfloor$. This confirms a conjecture attributed to Nelder (1979) and made independently by Mahmoodian and by Bate and van Rees. Hatami and Qian had shown that this minimum is at least $n^2/10^4$ for large $n$. The proof is a counting argument modulo two that uses only expansion of products, cancellation in pairs and the pigeonhole principle, and it is illustrated with partial Latin squares of orders three to seven. An appendix discloses the substantial use of AI in developing the proof and writing the exposition.