Matroid matching with Dilworth truncation

AbstractLet H=(V,E) be a hypergraph and let k⩾1 and l⩾0 be fixed integers. Let M be the matroid with ground-set E s.t. a set F⊆E is independent if and only if each X⊆V with k|X|-l⩾0 spans at most k|X|-l hyperedges of F. We prove that if H is dense enough, then M satisfies the double circuit property, thus Lovász’ min–max formula on the maximum matroid matching holds for M. Our result implies the Berge–Tutte formula on the maximum matching of graphs (k=1, l=0), generalizes Lovász’ graphic matroid (cycle matroid) matching formula to hypergraphs (k=l=1) and gives a min–max formula for the maximum matroid matching in the two-dimensional rigidity matroid (k=2, l=3)