THE SPECTRUM OF PBD ((5, K-ASTERISK), UPSILON) FOR K = 9, 13

A pairwise balanced design (PBD) of index 1 is a pair (V, A) where V is a finite set of points and A is a set of subsets (called blocks) of V, each of cardinality at least two, such that every pair of distinct points of V is contained in exactly one block of A. We may further restrict this definition to allow precisely one block of a given size, and in this case the design is called a PBD ({K, k*}, v) where k is the unique block size, K is the set of other allowable block sizes, and v is the number of points in the design. It is shown here that a PBD ({5, 9*}, v) exists for all v = 9 or 17 mod 20, v greater-than-or-equal-to 37, with the possible exception of 49, and that a PBD ({5, 13*}, v) exists for all v = 13 mod 20, v greater-than-or-equal-to 53.