Vertex-primitive s-arc-transitive digraphs of linear groups

We study G -vertex-primitive and ( G , s ) -arc-transitive digraphs for almost simple groups G with socle PSL n ( q ) . We prove that s ⩽ 2 for such digraphs, which provides the first step in determining an upper bound on s for all the vertex-primitive s -arc-transitive digraphs.