On the Number of Path Systems

A path system in a graph $G$ is a collection of paths, with exactly one path between any two vertices in $G$. A path system is said to be consistent if it is intersection-closed. We show that the number of consistent path systems on $n$ vertices is $n^{\frac{n^2}{2}(1-o(1))}$, whereas the number of consistent path systems which are realizable as the unique geodesics w.r.t. some metric is only $2^{\mathrm{\Theta }(n^2)}$. In addition, these insights allow us to improve known bounds on the face-count of the metric cone and shed new light on enumerating maximum-VC-classes.