Optimal spreadness for Latin-square distributions

Determine whether there exists a spread distribution over Latin squares of order n with spread parameter within a factor of two of the optimal value, in particular an approximately ((1+ε)e^2/n)-spread distribution, as required to avoid large subsquares by spreadness alone.

Background

The paper constructs an n{-1+γ}-spread distribution over high-girth, subsquare-free Latin squares, but this spreadness is insufficient to rule out large subsquares directly. The authors explain that avoiding all large subsquares by a union-bound argument would require a distribution with spread parameter approximately (1+ε)e2/n. They note that even the corresponding problem without the high-girth restriction is unresolved, while the high-girth version would be stronger still.

References

This level of spreadness is already an open problem for general Latin squares, while we would further need it for the family of high girth Latin squares.

— A Combinatorial Proof of Hilton's Conjecture and Beyond  (2609.28966 - Lesgourgues et al., 24 Sep 2026) in Section 1, subsection “A More General Theorem”; also Section 2, subsection “Small, Medium, and Large Subsquares”

Kelly further conjectured (see the comment after Corollary~1.7) that the {\em uniform} distribution over Latin squares of order $n$ is $((e2 +o(1))/n)$-spread.

— A Combinatorial Proof of Hilton's Conjecture and Beyond  (2609.28966 - Lesgourgues et al., 24 Sep 2026) in Section 1, subsection “A Highly Spread Existence Theorem”

We reiterate that the existence of a $\frac{(1+\varepsilon)e2}{n}$-spread distribution over high-girth Latin squares would be sufficient to prove the existence of subsquare-free Latin squares, by simple modifications of the proof of~\cref{claim:no_medium}. In light of this, we conjecture the existence of the following distribution.

— A Combinatorial Proof of Hilton's Conjecture and Beyond  (2609.28966 - Lesgourgues et al., 24 Sep 2026) in Section 2, immediately after the proof of Theorem 1.3