A theorem on the maximum number of disjoint steiner triple systems

We prove that D(2v + 1) ⩾ v + 1 + D(v) for v > 3 where D(v) denotes the maximum number of pairwise disjoint Steiner triple systems of order v. Since D(v) ⩽ v − 2 it follows that for v > 3, D(2v + 1) = 2v −1 whenever D(v) = v − 2.