The number of common flowers of two STS(v)STS(v)s and embeddable Steiner triple trades

A flower, FS(x)FS(x), around a point xx in a Steiner triple system D=(V,B)D=(V,B) is the set of all triples in BB which contain the point xx, namely FD(x)={b∈B∣x∈b}FD(x)={b∈B∣x∈b}. This paper determines the possible number of common flowers that two Steiner triple systems can have in common. For all admissible pairs (k,v)(k,v) where k≤v−6k≤v−6 we construct a pair of Steiner triple systems of order vv where the flowers around kk elements of VV are identical in both Steiner triple systems, except for the pairs (2,9)(2,9), (3,9)(3,9) and (6,13)(6,13). Equivalently this result shows that there is a Steiner triple trade of foundation l=v−kl=v−k that can be embedded in a STS(v)STS(v) for each admissible vv and 6≤l≤v6≤l≤v except when (l,v)=(6,9),(7,9)(l,v)=(6,9),(7,9) or (7,13)(7,13).