10.18453/rosdok_id00005899 Ge, Gennian Gennian Ge Grüttmüller, Martin Martin Grüttmüller http://d-nb.info/gnd/122522559 Hartmann, Sven Sven Hartmann http://d-nb.info/gnd/118124978 Rees, Rolf S. Rolf S. Rees http://d-nb.info/gnd/122169727 University of Rostock, Institute of Mathematics 2005 510 Mathematics 2005 en https://purl.uni-rostock.de/rosdok/id00005899 urn:nbn:de:gbv:28-rosdok_id00005899-8 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 necessery 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 ≥ 18k^2, or v ≡ 0 mod 3 and v ≥ 12k^3, or v ≡ 1 mod 3 and v ≥ 8k^4.