Decomposition of large uniform hypergraphs

We determine a minimum cardinality family _{n, k} (resp. _{n, k} ) ofn-uniform,k-edge hypergraphs satisfying the following property: all, except for finitely many,n-uniform hypergraphs satisfying the divisibility condition have an _{n, k} -decomposition (resp. vertex _{n, k} -decomposition).