The intersection problem for PBD(4, 7∗)1

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 (X, B_x) and (X, B₂) of order v containing the same block B of size 7 such that \ (B1/{B}) [B2/{B}\ = k. We will denote the set of all such k by J(v). I(v) = {0,1,..., b_v-8,b_v - 6,b_v}, where bv = {v²-v-42)/12 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 {10,19,22,31,34,46,58,70}. © 2018 Utilitas Mathematica Publishing Inc. All rights reserved.