The intersection problem for PBD(4,7)

Abstract

For every v = 7,10 (mod 12) with v > 22 there exist a pairwise balanced design PBD of order v with exactly one block of size 7 and rest of size 4, denoted by PBD(4,7) of order v. The intersection problem for PBD(4,7) is the determination of all pairs (v, k) such that there exist a pair of PBD(4,7)s (Xy B) and (X, #2) of order v containing the same block B of size 7 such that {B {B}) n {£})| = k. We will denote the set of all such k by J(v). I(v) = {0,1,... X-8,&t>-6,6V}, where bv = (v2-v-42)2 be the number of blocks of size 4 in PBD(4,7) of order v. It is established that J(v) = I(v) for any positive integer v = 7,10 (mod 12) and v i {10,19,22,31,34,46,58,70}. © 2019 Utilitas Mathematica Publishing Inc.. All rights reserved.