Separating path systems of almost linear size

A separating path system for a graph G G is a collection P \mathcal {P} of paths in G G such that for every two edges e e and f f , there is a path in P \mathcal {P} that contains e e but not f f . We show that every n n -vertex graph has a separating path system of size O ( n log ∗ ⁡ n ) O(n \log ^* n) . This improves upon the previous best upper bound of O ( n log ⁡ n ) 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 ) O(n) bound should hold.