Minimum embedding of Steiner triple systems into (K4-e)(K4-e)-designs I

A (K4-e)(K4-e)-design on v+wv+w points embeds   a Steiner triple system (STS) if there is a subset of vv points on which the graphs of the design induce the blocks of a STS. It is established that w⩾v/3w⩾v/3, and that when equality is met that such a minimum embedding of an STS(v)STS(v) exists, except when v=15v=15.