Small embeddings for partial triple systems of odd index

It has been conjectured that any partial triple system of order u and index λ can be embedded in a triple system of order v and index λ whenever v ⩾ 2 u + 1 , λ ( v − 1 ) is even and λ ( v 2 ) ≡ 0 ( mod 3 ) . This conjecture is known to hold for λ = 1 and for all even λ ⩾ 2 . Here the conjecture is proven for all remaining values of λ when u ⩾ 28 .