Mandatory Representation Designs MRD ( 4 , k ; v ) with k ≡ 2

A mandatory representation design MRD(K; v) is a pairwise balanced design on v points with block sizes from the set K in which for each k ∈ K there is at least one block in the design of size k. In this paper, we show that the necessary criteria for an MRD(K; v) to exist are asymptotically sufficient for finite K. Furthermore, we consider MRDs with K = {4, k}, where k ≡ 2 mod 3, k ≥ 5. Here, we prove that the necessary conditions for existence are sufficient if v ≡ 2 mod 3 and v ≥ 18k2, or v ≡ 0 mod 3 and v ≥ 12k3, or v ≡ 1 mod 3 and v ≥ 8k4.