Quasi-isometry between bounded-sphere-dimension graphs and ball intersection graphs

Establish that, for every integer d\geq 2, the class of graphs with sphere dimension at most d is quasi-isometric to the class of intersection graphs of balls in \mathbb{R}^d.

Background

The proposed quasi-isometry would relate sphere intersection graphs to intersection graphs of balls in the same Euclidean dimension. Such a relationship could potentially reduce the conjectured asymptotic-dimension bound for sphere graphs to known dimension bounds for ball intersection graphs.

The paper states that this reduction is known in dimension d=2, but presents the general assertion for d\geq 2 as a conjecture.

References

For any integer $d\geq 2$, the class of graphs with sphere dimension at most $d$ is quasi-isometric to the class of intersection graphs of balls in $\mathbb{R}d$.

— 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:qi}