On $t$-Multi-Set Designs

Abstract

A multi-set design of order $v$, ${MB}_t(v,k,\lambda )$, first defined by Assaf, Hartman, and Mendelsohn, is an ordered pair $(V,B)$, where $V$ is a set of cardinality $v$ and $B$ is a collection of multi-subsets of $V$ of size $k$ (called blocks), with the property that every multi-subset of $V$ of size $t$ is contained a total of $\lambda$ times in the blocks of $B$. (For example, the multi-set $\{a,b,b\}$ is contained $(\genfrac{}{}{0ex}{}{2}{1})(\genfrac{}{}{0ex}{}{4}{2})=12$ times in the multi-set $\{a,a,b,b,b,b,c\}$ and not at all in the multi-set $\{a,a,b,c\}$.) Previously, the first author had pointed out that any $t$-multi-set design is a $1$-design. Here, we show the pleasant yet not obvious fact that any $t$-multi-set design is a $t^{\prime }$-multi-set design for any positive integer $t^{\prime }\le t$.