Separating paths systems of almost linear size

A separating path system for a graph $G$ is a collection $\mathcal{P}$ of paths in $G$ such that for every two edges $e$ and $f$ in $G$, there is a path in $\mathcal{P}$ that contains $e$ but not $f$. We show that every $n$-vertex graph has a separating path system of size $O(n{\log }^{\ast }n)$. This improves upon the previous best upper bound of $O(n\log n)$, and makes progress towards a conjecture of Falgas-Ravry–Kittipassorn–Korándi–Letzter–Narayanan and Balogh–Csaba–Martin–Pluhár, according to which an $O(n)$ bound should hold.