Asymptotic gap between the two thresholds

Determine, for each fixed dimension d, whether NE_d(n) − NC_d(n) remains bounded and whether NE_d(n)/NC_d(n) remains bounded as n tends to infinity.

Background

The relationship between the extendibility and completability thresholds is largely unresolved. The authors explicitly note that the size of the gap is unknown and ask whether either the additive or multiplicative separation can grow without bound.

References

It is quite possible that the gap between these functions grows large, but this is not known. Question 9. For each d, is NEd (n)−NCd (n) bounded as n → ∞? Is NEd (n)/ NCd (n)bounded?

— Extendibility of Latin Hypercuboids  (2502.08868 - Bowtell et al., 13 Feb 2025) in Question 9, Section 5.2, p. 9

Determine the asymptotic behavior of the extension ratio

\frac{E(n)}{N(n)}.

In particular, is it true that

\frac{E(n)}{N(n)}\longrightarrow 0 \qquad (n\to\infty)?

More generally, determine the correct order of magnitude of E(n)/N(n).

— Counting Survivor Sets: Exponential Equivalence with Prime-Admissible Sets  (2609.08528 - Raso et al., 8 Sep 2026) in Section 10, “Open problems,” item 1