Independence Numbers of Random Subgraphs of Some Distance Graph

The distance graph G(n, 2, 1) is a graph where vertices are identified with two-element subsets of {1, 2, ... , n}, and two vertices are connected by an edge whenever the corresponding subsets have exactly one common element. A random subgraph G(p)(n, 2, 1) in the Erdos-Renyi model is obtained by selecting each edge of G(n, 2, 1) with probability p independently of other edges. We find a lower bound on the independence number of the random subgraph G(1/2)(n, 2, 1).