Nestings of BIBDs with block size four

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 (v,k,λ)-BIBD in such a way that we end up with a partial (v,k+1,λ+1)-BIBD. In the case where the partial (v,k+1,λ+1)-BIBD is in fact a (v,k+1,λ+1)-BIBD, we have a perfect nesting. We show that a nesting is perfect if and only if k = 2 λ + 1. Perfect nestings were previously known to exist in the case of Steiner triple systems (i.e., $(v,3,1)$-BIBDs) when v ≡ 1 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 ≡ 1 or 4 12, v ≥ 13. This is accomplished by a variety of direct and recursive constructions.