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 \emph{nested point}) to every block of a $(v,k,\lambda )$-BIBD in such a way that we end up with a partial $(v,k+1,\lambda +1)$-BIBD. In the case where the partial $(v,k+1,\lambda +1)$-BIBD is in fact a $(v,k+1,\lambda +1)$-BIBD, we have a \emph{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., $(v,3,1)$-BIBDs) when $v\equiv 1\mathrm{mod}6$, as well as for some symmetric BIBDs. Here we study nestings of $(v,4,1)$-BIBDs, which are not perfect nestings. We prove that there is a nested $(v,4,1)$-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.