Packing of partial designs

Abstract

We say that two hypergraphsH ₁ andH ₂ withv vertices eachcan be packed if there are edge disjoint hypergraphsH ^′₁ andH ^′₂ on the same setV ofv vertices, whereH ^′_i is isomorphic toH _i.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(v ^k−2t+1logv). Both results are nearly optimal asymptotically and answer questions of Teirlinck. The proofs are probabilistic.