Optimal asymptotic dimension for bounded sphere dimension

Prove that, for every integer d\geq 1, the class of graphs with sphere dimension at most d has asymptotic dimension at most d.

Background

The paper establishes that graphs with sphere dimension at most d have asymptotic dimension at most 2d+2, via a more general theorem for intersection graphs of path-connected sets obtained from bounded-aspect-ratio convex sets by deleting interior subsets. The authors conjecture that the optimal upper bound is d itself.

The conjectured value would be tight because the d-dimensional hypercubic lattice \mathbb{Z}d has both asymptotic dimension and sphere dimension equal to d. The paper notes that the cases d=1 and d=2 have subsequently available or forthcoming supporting results, while a general proof remains unresolved in the stated text.

References

For any integer $d\geq 1$, the class of graphs with sphere dimension at most $d$ has asymptotic dimension at most $d$.

— Strongly sublinear separators and bounded asymptotic dimension for sphere intersection graphs  (2504.00932 - Davies et al., 1 Apr 2025) in Section 6, “Final remarks and open problems,” Conjecture \ref{conj:asdim}