Small intersection numbers in the curve graph

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p\ge 0$ punctures, and let $\omega (g,p)=3g+p-4$. We prove the existence of infinitely long geodesic rays $\{v_0,v_1,v_2,\mathrm{.}\mathrm{.}\mathrm{.}\}$ in the curve graph satisfying the following optimal intersection property: for any natural number $k$, the endpoints $v_i,v_{i+k}$ of any length $k$ subsegment intersect $O({\omega }^{k-2})$ times. By combining this with work of the first author, we answer a question of Dan Margalit.