On the Maximal Number of Strongly Independent Vertices in a Random Acyclic Directed Graph

Let A𝑛 denote a random acyclic directed graph which is obtained from a random graph with vertex set {1,2,⋯,𝑛}, such that each edge is present with a prescribed probability p and all the edges are directed from higher to lower indexed vertices. Define a subset of vertices in A𝑛 to be strongly independent if there is no directed path between any pair of vertices in the subset. We show that the sequence J⁡(A𝑛), the number of vertices in the largest strongly independent vertex subset of A𝑛 satisfies with probability tending to 1 [...]