Pairwise balanced designs on 4s+1 points with longest block of cardinality 2s

Abstract

The quantity g^(k)(v) was introduced in [3] as the minimum number of blocks necessary in a pairwise balanced design on v elements, subject to the condition that the longest block have cardinality k. When k ≥ (v - 1)/2, except for the case where v ≡ 1 (mod 4) and k = (v - 1)/2, it is known that g^(k)(v) = 1 + (v - k)(3k - v + 1)/2. The designs which achieve this bound contain, apart from the long block, only pairs and triples, all of which intersect the long block. This paper investigates the exceptional case where v ≡ 1(mod 4) and k = (v - 1)/2. We prove that PBD(v)s with g^(k)(v) blocks contain, apart from the long block, only pairs, triples, and quadruples, all of which intersect the long block. We also give a comprehensive description for the structure of the PBD(v)s.