On the strength of connectedness of a random hypergraph

Bollobás and Thomason (1985) proved that for each k = k(n) ∈ [1, n − 1], with high probability, the random graph process, where edges are added to vertex set V = [n] uniformly at random one after another, is such that the stopping time of having minimal degree k is equal to the stopping time of becoming k-(vertex-)connected. We extend this result to the d-uniform random hypergraph process, where k and d are fixed. Consequently, for m = nd (lnn + (k − 1) ln lnn + c) and p = (d − 1)! lnn+(k−1) ln lnn+c nd−1, the probability that the random hypergraph models Hd(n,m) and Hd(n, p) are k-connected tends to e −e−c/(k−1)!