Packing of partial designs
Two hypergraphs \(H_ 1\) and \(H_ 2\) each with \(v\) vertices can be packed if there are edge disjoint hypergraphs \(H_ 1'\) and \(H_ 2'\) on the same set \(V\) of vertices where \(H_ i'\) is isomorphic to \(H_ i\). It was proven by \textit{B. Ganter, J. Pelikan} and \textit{L. Teirlinck} [Ars Comb. 4, 133-142 (1977; Zbl 0418.05003)] that if \(k\geq 2t\), then every two \((t,k,v)\) partial designs can be packed. In this paper it is shown that for any fixed integers \(k\) and \(t\) with \(t\leq k\leq 2t-2\) and for all sufficiently large \(v\), there are two \((t,k,v)\) partial designs that cannot be packed. Moreover, there are two isomorphic partial \((t,k,v)\)- designs that cannot be packed. It is also shown that for every fixed \(k\geq 2t-1\) and for all sufficiently large \(v\), there is a \((\lambda_ 1,t,k,v)\) partial design and a \((\lambda_ 2,t,k,v)\) partial design that cannot be packed, where \(\lambda_ 1\lambda_ 2\leq O(v^{k-2t+1}\log v)\). Both results are nearly optimal asymptotically and answer questions of Teirlinck. The proofs are probabilistic.
scientific article; zbMATH DE number 567479