Embedding Steiner triple systems in hexagon triple systems

A hexagon triple is the graph consisting of the three triangles (triples) { a , b , c } , { c , d , e } , and { e , f , a } , where a , b , c , d , e , and f are distinct. The triple { a , c , e } is called an inside triple. A hexagon triple system of order n is a pair ( X , H ) where H is a collection of edge disjoint hexagon triples which partitions the edge set of K n with vertex set X . The inside triples form a partial Steiner triple system. We show that any Steiner triple system of order n can be embedded in the inside triples of a hexagon triple system of order approximately 3 n .