Solution of two fractional packing problems of lovász

Abstract

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