  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 $\left\{v_{0},v_{1}, v_{2}, ...\right\}$ 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.
