Coloring numbers of touching graphs of unit balls

Determine the asymptotic behavior of the strong coloring number col_d for touching graphs of unit balls in R^d, improving the known bounds O(k^{d-1}) and Ω(k^{d/2}).

Background

The concluding remarks list an explicit question attributed to Dvořák et al. for touching graphs of unit balls in Euclidean space. The paper records the current upper and lower asymptotic bounds but does not determine the exact growth rate.

References

Dvo\v{r}ak et al.: What is the asymptotic behavior of $\col_d$ of touching graphs of unit balls in $\Rd$? It is known to be in $O(k{d-1})$ and $\Omega(k{d/2})$.

— On the generalized coloring numbers  (2501.08698 - Siebertz, 15 Jan 2025) in Section 10, Concluding remarks and acknowledgments