Paths and stability number in digraphs

The Gallai-Milgram theorem says that the vertex set of any digraph with stability number k can be partitioned into k directed paths. In 1990, Hahn and Jackson conjectured that this theorem is best possible in the following strong sense. For each positive integer k, there is a digraph D with stability number k such that deleting the vertices of any k-1 directed paths in D leaves a digraph with stability number k. In this note, we prove this conjecture.