Embedding partial steiner triple systems is NP-complete

It is known that any partial Steiner triple system of order υ can be embedded in a Steiner triple system of order w whenever w≥4υ+1, and w≡1, 3 (mod 6); moreover, it is conjectured that the same is true whenever w≥2υ+1. By way of contrast, it is proved that deciding whether a partial Steiner triple system of order υ can be embedded in a Steiner triple system of order w for any w≤2υ-1 is NP-complete. In so doing, it is proved that deciding whether a partial commutative quasigroup can be completed to a commutative quasigroup is NP-complete.