OPEN
This is open, and cannot be resolved with a finite computation.
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.
View the LaTeX source
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #724, https://www.erdosproblems.com/724, accessed 2026-01-16
