Maximum asymptotic gap between Euclidean and hyperbolic sphericity

Characterize the largest possible asymptotic difference between the Euclidean sphericity and the hyperbolic sphericity of a graph, and determine whether the logarithmic gap exhibited by stars is asymptotically maximal.

Background

The paper establishes comparisons between Euclidean and hyperbolic sphericity and identifies stars as a natural example with a logarithmic dimensional advantage in hyperbolic space. However, it does not determine whether any graph can exhibit a larger asymptotic separation. The unresolved issue is therefore to establish the maximum possible Euclidean-versus-hyperbolic sphericity gap.

References

Another open question is how large the difference between Euclidean and hyperbolic sphericity can become asymptotically. Here, too, stars are a canonical candidate. In this case, an EUBG representation requires at most logarithmically more dimensions than a HUBG representation. It is not obvious, whether this is the largest possible gap between Euclidean and hyperbolic sphericity.

— Hyperbolic Sphericity  (2610.01336 - Bläsius et al., 1 Oct 2026) in Section 5, Open Problems