The \emph{sphere dimension} of a graph $G$ is the smallest integer $d \geq 2$ so that $G$ is an intersection graph of metric spheres in $\R^d$. This talk considers the class $\mathcal{C}^{d}$ of graphs with sphere
dimension $d$. We present the result that for each integer $t$, the class of all graphs in $\mathcal{C}^{d}$ that exclude $K_{t,t}$ as a subgraph has strongly sublinear separators.
The presented work is joined with James Davies, Agelos Georgakopoulos and Rose McCarty.