On a phase transition of the random intersection graph: Supercritical region

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karo\'nski, Scheinerman and Singer-Cohen [Combin Probab Comput 8 (1999), 131--159] introduced a random intersection graph by taking random assigned sets. The random intersection graph $G(n,m;p)$ has $n$ vertices and their assigned sets are chosen to be i.i.d. random subsets of a fixed set $M$ of size $m$ where each element of $M$ belongs to each random subset with probability $p$, independently of all other elements in $M$. Fill, Scheinerman and Singer-Cohen [Random Struct Algorithms 16 (2000), 156--176] showed that the total variation between the random graph $G(n,m;p)$ and the Erd\"os-R\'enyi graph $G(n,\hat{p})$ tends to $0$ if $m=n^{\alpha}$, $\alpha >6$, where $\hat{p}$ is chosen so that the expected numbers of edges in the two graphs are the same. In this paper, it is proved that the total variation still tends to $0$ whenever $m \gg n^4$. We believe that this is the best possible.