Solution of two fractional packing problems of Lovász

Lovász asked whether the following is true for each hypergraph H and natural number k: (*) if vk (H') = k · v* (H') holds for each hypergraph H' arising from H by multiplication of points, then vk(H) = tk(H); (**) if tk(H') = k · t*(H') holds for each hypergraph H' arising by removing edges, then tk (H) = vk (H). We prove and generalize assertion (*) and give a counterexample to (**).