Linear lower bound for hyperbolic sphericity of complete bipartite graphs

Determine whether the hyperbolic sphericity of the complete bipartite graph K_{n,n} requires a number of dimensions linear in n, analogous to the known Euclidean and dot-product lower bounds.

Background

The paper asks whether hyperbolic sphericity can grow linearly with the number of vertices. In Euclidean and dot-product representations, the complete bipartite graph K_{n,n} requires at least n dimensions. The authors explain that the Euclidean proof does not transfer readily to hyperbolic geometry and that only a logarithmic lower bound is currently available through results for non-uniform ball graphs, which generalize hyperbolic unit ball graphs.

References

It remains unclear whether the hyperbolic sphericity can grow linearly with the size of the graph. For Euclidean or dot-product representations, it is known that the complete bipartite graph $K_{n,n}$ requires at least $n$ dimensions. We ask whether a similar requirement holds in the hyperbolic case.

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