Complexity of representation of graphs by set systems

Let $\text{F}$ be a family of subsets of S and let G be a graph with vertex set $\text{V={x}{}_{\mathrm{A}}\text{|A ∈}\text{F}\text{}}$ such that: (x_A, x_B) is an edge iff $\text{A?B≠}\text{0}\text{/}$. The family $\text{F}$ is called a set representation of the graph G.It is proved that the problem of finding minimum k such that G can be represented by a family of sets of cardinality at most k is NP-complete. Moreover, it is NP-complete to decide whether a graph can be represented by a family of distinct 3-element sets.The set representations of random graphs are also considered.