Nestings of BIBDs with block size four

Abstract

In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the nested point) to every block of a $\left(v,k,\lambda \right)$-BIBD in such a way that we end up with a partial $\left(v,k+1,\lambda +1\right)$-BIBD. In the case where the partial $\left(v,k+1,\lambda +1\right)$-BIBD is in fact a $\left(v,k+1,\lambda +1\right)$-BIBD, we have a perfect nesting. We show that a nesting is perfect if and only if $k=2\lambda +1$. Perfect nestings were previously known to exist in the case of Steiner triple systems (i.e., $\left(v,3,1\right)$-BIBDs) when $v\equiv 1\mathrm{mod}6$, as well as for some symmetric BIBDs. Here we study nestings of $\left(v,4,1\right)$-BIBDs, which are not perfect nestings. We prove that there is a nested $\left(v,4,1\right)$-BIBD if and only if $v\equiv 1\text{or}4\mathrm{mod}12$, $v\ge 13$. This is accomplished by a variety of direct and recursive constructions.