**Gennian Ge, Martin Grüttmüller, Sven Hartmann, Rolf Rees**

**Mandatory Representation Designs MRD$(4,k;v)$ with $k\equiv 2 \mod 3$**

**Preprint series:**
Preprints aus dem Fachbereich Mathematik, Universität Rostock

**MSC:**- 05B30 Other designs, configurations, See also {51E30}

**Abstract:** 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 $k\in K$ 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 $k\equiv 2\mod 3, k\geq 5$. Here, we prove that the necessary conditions for existence are sufficient if $v\equiv 2\mod 3$ and $v\geq 18k^2$, or $v\equiv 0\mod 3$ and $v\geq 12k^3$, or $v\equiv 1\mod 3$ and $v\geq 8k^4$.

**Keywords:** *pairwise balanced design, mandatory representation design, asymptotic existence*