k-connectivity of uniform s-intersection graphs

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