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 upsilon points, index one, blocks of size 3 and 5, and r(3), r(5) > 0, are upsilon equivalent to 15 (mod 30). If r(k) > 1, then upsilon > k(2), and r(3) = (upsilon - 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 upsilon = 105, and for r(5) = 2, 4, 5 with possible exceptions (upsilon = 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. (C) 2009 Elsevier B.V. All rights reserved.