On small subgraphs in a random intersection digraph

Given a set of vertices V and a set of attributes W let each vertex v ∈ V include an attribute w ∈ W into a set S − ( v ) with probability p − and let it include w into a set S + ( v ) with probability p + independently for each w ∈ W . The random binomial intersection digraph on the vertex set V is defined as follows: for each u , v ∈ V the arc u v is present if S − ( u ) and S + ( v ) are not disjoint. For any h = 2 , 3 , … we determine the birth threshold of the complete digraph on h vertices and describe the configurations of intersecting sets that realise the threshold.