Pairwise Balanced Designs on \(4s + 4\) blocks with longest block of cardinality \(2s + 1\)

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 \(k \geq (v – 1)/2\), it is known that \(g^{(k)}(v) = 1 + (v – k)(3k – v + 1)/2\), except for the case when \(v \equiv 1 \pmod{4}\) 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 \(v = 4s + 4\).