Separating the edges of a graph by a linear number of paths

Recently, Letzter proved that any graph of order n contains a collection P of O(nlog⋆n) paths with the following property: for all distinct edges e and f there exists a path in P which contains e but not f . We improve this upper bound to 19n, thus answering a question of G.O.H. Katona and confirming a conjecture independently posed by Balogh, Csaba, Martin, and Pluhár and by Falgas-Ravry, Kittipassorn, Korándi, Letzter, and Narayanan. Our proof is elementary and self-contained.