Conformal Hypergraph
The primal graph G(H) of a hypergraph is the graph on the same vertex set that has as its edges the pairs of vertices appearing together in the same hyperedge. A hypergraph is said to be conformal if every maximal clique of its primal graph is a hyperedge, or equivalently, if every clique of its primal graph is contained in some hyperedge. If the hypergraph is required to be downward-closed (so it contains all hyperedges that are contained in some hyperedge) then the hypergraph is conformal precisely when it is a flag complex. This relates the language of hypergraphs to the language of simplicial complexes.
Read more about this topic: Clique Complex