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 kk (denoted kk-pc). The number of kk-pcs is denoted rkrk. The necessary conditions for URDs with vv points, index one, blocks of size 3 and 5, and r3,r5>0r3,r5>0, are v≡15(mod30). If rk>1rk>1, then v≥k2v≥k2, and r3=(v−1−4⋅r5)/2r3=(v−1−4⋅r5)/2. For r5=1r5=1 these URDs are known as group divisible designs. We prove that these necessary conditions are sufficient for r5=3r5=3 except possibly v=105v=105, and for r5=2,4,5r5=2,4,5 with possible exceptions (v=105,165,285,345v=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.