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 yz are already in the graph and xyz∈H. We say that the process propagates if all the edges are added to the graph eventually. In this paper we prove that the threshold probability for propagation is p=1/2√n. We also show that p=1/2√n is an upper bound for the threshold probability that a random 2-dimensional simplicial complex is simply-connected.