Independence numbers of random subgraphs of a distance graph

We consider the so-called distance graph G(n, 3, 1), whose vertices can be identified with three-element subsets of the set {1, 2,..., n}, two vertices being joined by an edge if and only if the corresponding subsets have exactly one common element. We study some properties of random subgraphs of G(n, 3, 1) in the Erdős–Rényi model, in which each edge is included in the subgraph with some given probability p independently of the other edges. We find the asymptotics of the independence number of a random subgraph of G(n, 3, 1).