Allston, J. L.; Grannell, M. J.; Griggs, T. S. and Stanton, R. G.
Pairwise balanced designs on points with longest block of cardinality .
Utilitas Mathematica, 58, pp. 97–107.
The quantity is the minimum number of blocks necessary in a pairwise balanced design on elements, subject to the condition that the longest block have cardinality . When , except for the case where (mod 4) and , it is known that . 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 (mod 4) and . We prove that s with 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
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