Uniformly resolvable designs with index one, block sizes three and five and up to five parallel classes with blocks of size five

Each parallel class of a uniformly resolvable design (URD) contains blocks of only one block size k (denoted k -pc). The number of k -pcs is denoted r k . The necessary conditions for URDs with v points, index one, blocks of size 3 and 5, and r 3 , r 5 > 0 , are v ≡ 15 ( mod 30 ) . If r k > 1 , then v ≥ k 2 , and r 3 = ( v − 1 − 4 ⋅ r 5 ) / 2 . For r 5 = 1 these URDs are known as group divisible designs. We prove that these necessary conditions are sufficient for r 5 = 3 except possibly v = 105 , and for r 5 = 2 , 4 , 5 with possible exceptions ( v = 105 , 165 , 285 , 345 ) New labeled frames and labeled URDs, which give new URDs as ingredient designs for recursive constructions, are the key in the proofs.