Pairwise Balanced Designs on $4s+4$ blocks with longest block of cardinality $2s+1$

Abstract

The quantity $g^{(k)}(v)$ was introduced in [4] 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 $k\ge (v–1)/2$, it is known that $g^{(k)}(v)=1+(v–k)(3k–v+1)/2$, except for the case when $v\equiv 1(\mathrm{mod}4)$ and $k=(v–1)/2$. This exceptional “case of first failure” was treated in [1] and [2]. In this paper, we discuss the structure of the “case of first failure” for the situation when $v=4s+4$.