# On the Number of Path Systems

## TL;DR

This paper investigates the enumeration of consistent path systems in graphs, revealing their exponential growth and distinguishing between general and metric realizable systems, with implications for metric cone face-counts and VC-classes.

## Contribution

It provides asymptotic counts for consistent path systems and differentiates those realizable as metric geodesics, advancing understanding in graph theory and metric geometry.

## Key findings

- Number of consistent path systems is approximately n^{n^2/2}
- Realizable metric path systems are exponentially fewer, about 2^{Θ(n^2)}
- Results improve bounds on metric cone face-count and VC-class enumeration.

## Abstract

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 $2^{\Theta(n^2)}$. 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.