Minimum embedding of Steiner triple systems into -designs II

A ( K 4 − e ) -design of order v + w embeds a given Steiner triple system if there is a subset of v points on which the graphs of the design induce the blocks of the original Steiner triple system. It has been established that w ≥ v / 3 , and that when equality is met, such a minimum embedding of an STS( v ) exists, except when v = 15 . Equality only holds when v ≡ 15 , 27 ( mod 30 ) . One natural question is: What is the smallest order w such that some STS ( v ) can be embedded into a ( K 4 − e ) -design of order v + w ? We solve the problem for 7 of the 10 congruence classes modulo 30.