Mandatory representation designs MRD(4,k;v) with k≡2mod3k≡2mod3

A mandatory representation design MRD(K;v) is a pairwise balanced design on vv points with block sizes from the set KK in which for each k∈Kk∈K there is at least one block in the design of size kk. In this paper, we consider MRDs with K={4,k}K={4,k}, where k≡2mod3,k≥5k≡2mod3,k≥5, and prove that the necessary conditions for existence are sufficient if v≡2mod3v≡2mod3 and v≥18k2v≥18k2, or v≡1mod3v≡1mod3 and v≥12k3v≥12k3, or v≡0mod3v≡0mod3 and v≥8k4v≥8k4.