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

Allston, J. L.; Grannell, M. J.; Griggs, T. S. and Stanton, R. G. (2000). Pairwise balanced designs on $4s+1$ points with longest block of cardinality $2s$. Utilitas Mathematica, 58, pp. 97–107.

Abstract

The quantity $g^{(k)}(v)$ is 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 $k \ge (v-1)/2$, except for the case where $v \equiv 1$ (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 $v \equiv 1$(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.

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