take a complete graph "k". it has 15 edges, 15 perfect matchings and 20 triangles. create a point for each of the 15 edges, and a line for each of the 20 triangles and 15 matchings. the incidence structure between each triangle or matching (line) to its three constituent edges (points), induces a pg(3,2). (sylvester's work on duads and synthemes, 1859)