Pairwise balanced designs on \(4s+1\) points with longest block of cardinality \(2s\) (Q2707952)

The quantity \(g^{(k)}(v)\) was introduced by \textit{R. G. Stanton, J. L. Allston} and \textit{D. D. Cowan} [Pair-coverings with restricted largest block length, Ars Comb. 11, 85-98 (1981; Zbl 0474.05022)] as the minimum number of blocks necessary in a pairwise balanced design (with \(\lambda = 1\)) on \(v\) elements (i.e. a \(\text{PBD}(v)\)), subject to the condition that the longest block have cardinality \(k\). \textit{D. R. Woodall} [The \(\lambda-\mu\) problem, J. Lond. Math. Soc., II. Ser. 1, 505-519 (1968; Zbl 0181.02102)] showed that \(g^{(k)}(v) \geq 1+(v-k)(3k-v+1)/2\). Later Stanton, Allston and Cowan showed (see above reference) that if \(v \equiv 1 \pmod{4}\) this bound is achieved for \(k > (v-1)/2\); otherwise they showed that the bound is achieved for \(k \geq (v-1)/2\). For \(v > 7\) the longest block is uniquely defined. Moreover, Stanton et al. showed that the only designs that achieve this bound contain, apart from the long block, only pairs and triples, all of which intersect the long block. \textit{R. C. Mullin, R. G. Stanton} and \textit{D. R. Stinson} [Perfect pair-coverings and an algorithm for certain (1-2) factorizations of the complete graph \(K_{2n+1}\), Ars Comb. 12, 73-80 (1981; Zbl 0476.05027)] considered the case where \(v = 4s+1\), \(k=2s\), \(s \geq 2\) and proved that \(g^{(2s)}(4s+1) = 2s^2+s+1+ \lceil s/2 \rceil\). The designs that were constructed to achieve this bound contain, apart from the long block which is unique if \(s>2\), only pairs, triples and quadruples. Also, all of these blocks intersect the long block. In this paper the authors consider this case further, describing designs which achieve this bound in more detail, and showing that they all have a similar structure. The authors conclude the paper by discussing the two smallest cases. For \(v=9\) and \(k=4\) the \(\text{PBD}(v)\) which achieves the minimum bound is unique up to isomorphism. For the next case, \(v=13\) and \(k=6\), the authors use the information they have obtained in this paper to enumerate the 9 designs that achieve the lower bound, thereby confirming the result of \textit{M. J. Grannell, T. S. Griggs, K. A. S. Quinn} and \textit{R. G. Stanton} [A census of minimal pair-coverings with restricted largest block length, Ars Comb. 52, 71-96 (1999)], who had obtained their enumeration by a more complex method.