# Pairwise balanced designs

Let $X$ be a finite set containing $v$ elements and $\lambda$ be a positive integer. Let $K$ be a set of positive integers. Further let there be a multiset $\mathcal{B}$ containing subsets of $X$ which are called blocks. The cardinality of each block is an element of $K$. The multiset $\mathcal{B}$ has the property that every pair-set $\{x,y\}\subset X$ is contained in exactly $\lambda$ blocks. We refer to the pair $(X,\mathcal{B})$ as a pairwise balanced design $B[K,\lambda;v]$.

Let $B(K,\lambda)=\{v:\exists B[K,\lambda;v]\}$. I wish to prove that $K\subset B(K,\lambda)$. It is said in the paper I am reading this from (H. Hanani (1975). "Balanced incomplete block designs and related designs") that this is self evident, but I do not find it so.

If $\lambda=1$ then to construct a $B[K,1;k]$ for any $k\in K$ we can take the set of $2$-subsets of $X=\{1,2\cdots k\}$ as $\mathcal{B}$ (and repeat each of them $\lambda$ times for a general $\lambda$), but again I am not even sure that $2\in K$.

Am I missing something obvious here?

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There will always exist some 'trivial' designs. For any $k \in K$, define $\mathcal{B}$ as containing $\lambda$ copies of $X$ ($X$ is of size $k$). Clearly each pair of elements is present in exactly $\lambda$ subsets and all subset sizes are contained in $K$.

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