linguistic:paraphrasing
All graphs are intersection graphs

Any undirected graph G may be represented as an intersection graph. For each vertex v of G, form a set S consisting of the edges incident to v; then two such sets have a nonempty intersection if and only if the corresponding vertices share an edge. Therefore, G is the intersection graph of the sets S.

Erdős, Goodman & Pósa (1966) provide a construction that is more efficient, in the sense that it requires a smaller total number of elements in all of the sets S combined. For it, the total number of set elements is at most n / 4, where n is the number of vertices in the graph. They credit the observation that all graphs are intersection graphs to Szpilrajn-Marczewski (1945), but say to see also Čulík (1964). The intersection number of a graph is the minimum total number of elements in any intersection representation of the graph.