Packing of partial designs

We say that two hypergraphs H1 and H2 with v vertices each can be packed if there are edge disjoint hypergraphs H ′ 1 and H ′ 2 on the same set V of v vertices, where H ′ i is isomorphic to Hi. It is shown that for every fixed integers k and t, where t ≤ k ≤ 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 ≥ 2t − 1 and for all sufficiently large v there is a (λ1, t, k, v) partial design and a (λ2, t, k, v) partial design that cannot be packed, where λ1λ2 ≤ O(vk−2t+1 log v). Both results are nearly optimal asymptotically and answer questions of Teirlinck. The proofs are probabilistic.