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}{a,b,c},{c,d,e}, and {e,f,a}{e,f,a}, where a,b,c,d,ea,b,c,d,e, and ff are distinct. The triple {a,c,e}{a,c,e} is called an inside triple. A hexagon triple system of order nn is a pair (X,H)(X,H) where HH is a collection of edge disjoint hexagon triples which partitions the edge set of KnKn with vertex set XX. The inside triples form a partial Steiner triple system. We show that any Steiner triple system of order nn can be embedded in the inside triples of a hexagon triple system of order approximately 3n3n.