On uniformly resolvable designs with block sizes 3 and 4

A Uniformly Resolvable Design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k, such a class is denoted k -pc and for a given k the number of k -pcs is denoted r _k . In this paper we consider the case of block sizes 3 and 4. The cases r ₃ = 1 and r ₄ = 1 correspond to Resolvable Group Divisible Designs (RGDD). We prove that if a 4-RGDD of type h ^u exists then all admissible {3, 4}-URDs with 12hu points exist. In particular, this gives existence for URD with v ≡ 0 (mod 48) points. We also investigate the case of URDs with a fixed number of k -pc. In particular, we show that URDs with r ₃ = 4 exist, and that those with r ₃ = 7, 10 exist, with 11 and 12 possible exceptions respectively, this covers all cases with 1 < r ₃ ≤ 10. Furthermore, we prove that URDs with r ₄ = 7 exist and that those with r ₄ = 9 exist, except when v = 12, 24 and possibly when v = 276. In addition, we prove that there exist 4-RGDDs of types 2 ¹⁴², 2 ³⁴⁶ and 6 ⁵⁴. Finally, we provide four {3,5}-URDs with 105 points.