The intersection problem for s(2,4, v)s with a common parallel class

Abstract

A parallel class in a design is a set of blocks that partition the point set. The intersection problem for Steiner systems with a common parallel class is the determination of all pairs (v, s) such that there exists a pair of Steiner systems (X, B) and (X, B2) of order v having a common parallel class V satisfying (B V)0(#2 V) = s. In this paper the intersection problem for a pair of S(2,4, u)'s with a common parallel class is investigated. Let J(u) = {s : 3 a pair of S(2,4,4u)'s intersecting in 3 + u blocks, u of them being the blocks of a common parallel class}; I(u) = {0,1,... ,pu-8,pu-6,pu}, where pu = 4ti(u-l)3 and pu + u is the number of blocks of an S(2,4,4u). It is established that J(u) = I(u) for any positive integer u = 1 (mod 3) and u 4,7,10,16,22,25. © 2020 Utilitas Mathematica Publishing Inc.. All rights reserved.