Solution of two fractional packing problems of Lovász

Lovász asked whether the following is true for each hypergraph H and natural number k:(**) if νk(H′)=k·ν*(H′)νk(H′)=k·ν*(H′) holds for each hypergraph H′H′ arising from H   by multiplication of points, then νk(H)=τk(H)νk(H)=τk(H);(****) if τk(H′)=k·τ*(H′)τk(H′)=k·τ*(H′) holds for each hypergraph H′H′ arising from H   by removing edges, then τk(H)=νk(H)τk(H)=νk(H).We prove and generalize assertion (**) and give a counterexample to (****).