The flower intersection problem for S(2,4,v)'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(2,4,3u+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=3u(u-1)/4 and f"u+u is the number of blocks of an S(2,4,3u+1). It is established that J(u)=I(u) for any positive integer u=0,1(mod4) and u5,8,9,12.