The flower intersection problem for ’s

A flower in a Steiner system is the set of all blocks containing a given point. The flower intersection problem for Steiner systems is the determination of all pairs ( v , s ) such that there exists a pair of Steiner systems ( X , B 1 ) and ( X , B 2 ) of order v having a common flower F satisfying | ( B 1 ∖ F ) ∩ ( B 2 ∖ F ) | = s . In this paper the flower intersection problem for a pair of S ( 2 , 4 , v ) ’s is investigated. Let J ( u ) = { s : ∃ a pair of S ( 2 , 4 , 3 u + 1 ) ’s intersecting in s + u blocks, u of them being the blocks of a common flower } . Let I ( u ) = { 0 , 1 , … , f u − 8 , f u − 6 , f u } , where f u = 3 u ( u − 1 ) / 4 and f u + u is the number of blocks of an S ( 2 , 4 , 3 u + 1 ) . It is established that J ( u ) = I ( u ) for any positive integer u ≡ 0 , 1 ( mod 4 ) and u ≠ 5 , 8 , 9 , 12 .