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

摘要

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-1) and (X, B-2) of order v having a common parallel class P satisfying vertical bar (B-1 P) boolean AND(B-2 P)vertical bar = s. In this paper the intersection problem for a pair of S(2, 4, v)'s with a common parallel class is investigated. Let J(u) = {s : there exists a pair of S(2, 4, 4u)'s intersecting in s + u blocks, u of them being the blocks of a common parallel class}; I(u) = {0, 1,...,P-u - 8, P-u - 6, P-u}, where p(u) = 4u(u - 1)/3 and p(u) + 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 equivalent to 1 (mod 3) and u not equal 4, 7, 10, 16, 22, 25.