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 $\{1,2,\dots,n\}$, 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 $G(n,3,1)$.