Curvature-free linear length bounds on geodesics in closed Riemannian surfaces

This paper proves that in any closed Riemannian surface $M$ with diameter $d$, the length of the $k^text{th}$-shortest geodesic between two given points $p$ and $q$ is at most $8kd$. This bound can be tightened further to $6kd$ if $p = q$. This improves prior estimates by A. Nabutovsky and R. Rotman.