Matroid matching with Dilworth truncation

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