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 . We also show that is an upper bound for the threshold probability that a random 2-dimensional simplicial complex is simply-connected.