The minimum number of blocks having maximum size precisely four that are required to cover, exactly ? times, all pairs of elements from a set of cardinality v is denoted by g? (4)(v) (or g(4)(v) when ? = 1). All values of g? (4) (v) are known except for ? = 1 and v = 17 or 18. It is known that 30)? g(4)(l7) ? 31 and 32 ? g(4)(l8) ? 33. In this paper we show that g(4)(17?30 and g(4)(18) ? 32, thus finalising the determination of g? (4)v) for all ? and v.
|Number of pages||11|
|Journal||Australasian Journal of Combinatorics|
|Publication status||Published - 2006|
Gruettmueller, M., Roberts, I., D'arcy, S., & Egan, J. (2006). The minimum number of blocks in pairwise balanced designs with maximum block size 4: the final cases. Australasian Journal of Combinatorics, 36, 303-313.