The asymptotic existence of BIBDs having a nesting

Abstract

A \((v,k,\lambda )\)-BIBD \((X,\mathcal {B})\) has a nesting if there is a mapping \(\phi :\mathcal {B}\rightarrow X\) such that \((X,\{B\cup \{\phi (B)\}\mid B\in \mathcal {B}\})\) is a \((v,k+1,\lambda +1)\)-packing. If the \((v,k+1,\lambda +1)\)-packing is a \((v,k+1,\lambda +1)\)-BIBD, then this nesting is said to be perfect. We show that given any positive integers k and \(\lambda \), if \(k\ge 2\lambda +2\), then for any sufficiently large v, every \((v,k,\lambda )\)-BIBD can be nested into a \((v,k+1,\lambda +1)\)-packing; and if \(k=2\lambda +1\), then for any sufficiently large v satisfying \(v \equiv 1 \pmod {2k}\), there exists a \((v,k,\lambda )\)-BIBD having a perfect nesting. Banff difference families (BDF), as a special kind of difference families (DF), can be used to generate nested BIBDs. We show that if G is a finite abelian group with a large size whose number of order 2 elements is no more than a given constant, and \(k\ge 2\lambda +2\), then one can obtain a \((G,k,\lambda )\)-BDF by taking any \((G,k,\lambda )\)-DF and then replacing each of its base blocks by a suitable translation. This is a Novák-like theorem. The generalized Novák’s conjecture states that given any positive integers k and \(\lambda \) with \(k\ge \lambda +1\), there exists an integer \(v_0\) such that, for any cyclic \((v,k,\lambda )\)-BIBD with \(v\ge v_0\), it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We confirm this conjecture for every \(k\ge \lambda +2\). Most of the theorems in this paper are based on a recent result presented by Delcourt and Postle on the asymptotic existence of an A-perfect matching of a bipartite hypergraph.