OPEN

Let $f(n)$ be the maximum number of mutually orthogonal Latin squares of order $n$. Is it true that
\[f(n) \gg n^{1/2}?\]

Euler conjectured that $f(n)=1$ when $n\equiv 2\pmod{4}$, but this was disproved by Bose, Parker, and Shrikhande [BPS60] who proved $f(n)\geq 2$ for $n\geq 7$.

Chowla, Erdős, and Straus [CES60] proved $f(n) \gg n^{1/91}$. Wilson [Wi74] proved $f(n) \gg n^{1/17}$. Beth [Be83c] proved $f(n) \gg n^{1/14.8}$.

The sequence of $f(n)$ is A001438 in the OEIS.