Packing of partial designs

We say that two hypergraphsH1 andH2 withv vertices eachcan be packed if there are edge disjoint hypergraphsH1′andH2′on the same setV ofv vertices, whereHi′is isomorphic toHi.It is shown that for every fixed integersk andt, wheret≤k≤2t−2 and for all sufficiently largev there are two (t, k, v) partial designs that cannot be packed. Moreover, there are twoisomorphic partial (t, k, v)-designs that cannot be packed. It is also shown that for every fixedk≥2t−1 and for all sufficiently largev there is a (λ1,t,k,v) partial design and a (λ1,t,k,v) partial design that cannot be packed, where λ1 λ2≤O(vk−2t+1logv). Both results are nearly optimal asymptotically and answer questions of Teirlinck. The proofs are probabilistic.