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)(v,s) such that there exists a pair of Steiner systems (X,B1)(X,B1) and (X,B2)(X,B2) of order vv having a common flower FF satisfying |(B1∖F)∩(B2∖F)|=s|(B1∖F)∩(B2∖F)|=s. In this paper the flower intersection problem for a pair of S(2,4,v)’s is investigated. Let J(u)={s:∃J(u)={s:∃ a pair of S(2,4,3u+1)’s intersecting in s+us+u blocks, uu of them being the blocks of a common flower}}. Let I(u)={0,1,…,fu−8,fu−6,fu}I(u)={0,1,…,fu−8,fu−6,fu}, where fu=3u(u−1)/4fu=3u(u−1)/4 and fu+ufu+u is the number of blocks of an S(2,4,3u+1). It is established that J(u)=I(u)J(u)=I(u) for any positive integer u≡0,1(mod4) and u≠5,8,9,12u≠5,8,9,12.