Weak and Strong Nestings of BIBDs

We study two types of nestings of balanced incomplete block designs (BIBDs). In both types of nesting, 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 (w,k+1,λ+1)-BIBD for some w ≥ v. In the case where $w > v$, we are introducing $w-v$ new points. This is called a weak nesting. A strong nesting satisfies the stronger property that no pair containing a new point occurs more than once in the partial (w,k+1,λ+1)-BIBD. In both cases, the goal is to minimize w. We prove lower bounds on w as a function of v, k and λ and we find infinite classes of $(v,2,1)$- and $(v,3,2)$-BIBDs that have optimal nestings.