The number of common flowers of two s and embeddable Steiner triple trades

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 ∈ 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 ) 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 l = v − k that can be embedded in a S T S ( v ) for each admissible v and 6 ≤ l ≤ v except when ( l , v ) = ( 6 , 9 ) , ( 7 , 9 ) or ( 7 , 13 ) .