Induced universal hypergraphs

We prove that the minimum number of vertices of a hypergraph that contains every d-uniform hypergraph on k vertices as an induced subhypergraph is (1 + o(1))2 ^{\Bigl(k d \Bigr) /k}. The proof relies on the probabilistic method and provides a nonconstructive solution. In addition we exhibit an explicit construction of a hypergraph on \Theta (2 ^{\Bigl(k d \Bigr) /k}) vertices, containing every d-uniform hypergraph on k vertices as an induced subhypergraph.