A property on reinforcing edge-disjoint spanning hypertrees in uniform hypergraphs

We prove that if H is a simple uniform hypergraph with |E(H)|=k(|V(H)|−1) and ε(H)>0, then there exist e∈E(H) and e′∈E(Hc) such that ε(H−e+e′)<ε(H). This generalizes a former result, which settles a conjecture of Payan. The result iteratively defines a finite ε-decreasing sequence of uniform hypergraphs H0,H1,H2,…,Hm such that H0=H, Hm is the union of k edge-disjoint spanning hypertrees, and such that two consecutive hypergraphs in the sequence differ by exactly one hyperedge.