The minimum number of blocks in pairwise balanced designs with maximum blocksize 4: the final cases

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)(17) 31 and 32 g(4)(18) 33. In this paper we show that g(4)(17) 6= 30 and g(4)(18) 6= 32, thus finalising the determination of g(4) (v) for all and v. 1