A random triadic process

Given a random 3-uniform hypergraph H = H(n, p) on n vertices where each triple independently appears with probability p, consider the following graph process. We start with the star G₀ on the same vertex set, containing all the edges incident to some vertex v₀, and repeatedly add an edge xy if there is a vertex z such that xz and zy are already in the graph and xzy∈ H. We say that the process propagates if it reaches the complete graph before it terminates. In this paper we prove that the threshold probability for propagation is p = 1/2√n. We conclude that p = 1 /2√n is an upper bound for the threshold probability that a random 2-dimensional simplicial complex is simply connected.