K-connectivity of Uniform S-intersection Graphs

Let W1, . . . ,Wn be independent random subsets of [m] = {1, . . . ,m}. Assuming that each Wi is uniformly distributed in the class of d-subsets of [m] we study the uniform random intersection graph Gs(n,m, d) on the vertex set {W1, . . .Wn}, defined by the adjacency relation: Wi ∼ Wj whenever |Wi ∩Wj | ≥ s. We show that as n,m → ∞ the edge density threshold for the property that each vertex of Gs(n,m, d) has at least k neighbours is asymptotically the same as that for Gs(n,m, d) being k-connected.