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

Abstract

A flower, F-s(x), around a point x in a Steiner triple system D = (V, B) is the set of all triples in B which contain the point x, namely F-D(X)={b is an element of B vertical bar x is an element of 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) where k <= v-6 we construct a pair of Steiner triple systems of order v where the flowers around k elements of V are identical in both Steiner triple systems, except for the pairs (2, 9), (3, 9) and (6, 13). Equivalently this result shows that there is a Steiner triple trade of foundation I = v k that can be embedded in a STS(v) for each admissible v and 6 <= l <= v except when (l, v) = (6, 9), (7, 9) or (7, 13).