kk-connectivity of uniform s-intersection graphs

Let W1,…,WnW1,…,Wn be independent random subsets of [m]={1,…,m}[m]={1,…,m}. Assuming that each WiWi is uniformly distributed in the class of dd-subsets of [m][m] we study the uniform random intersection graph Gs(n,m,d)Gs(n,m,d) on the vertex set {W1,…Wn}{W1,…Wn}, defined by the adjacency relation: Wi∼WjWi∼Wj whenever ∣Wi∩Wj∣≥s∣Wi∩Wj∣≥s. We show that as n,m→∞n,m→∞ the edge density threshold for the property that each vertex of Gs(n,m,d)Gs(n,m,d) has at least kk neighbours is asymptotically the same as that for Gs(n,m,d)Gs(n,m,d) being kk-connected.