Matroid matching with Dilworth truncation

Let H = (V, E) be a hypergraph and let k ≥ 1 and l ≥ 0 be fixed integers. LetM 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, thenM satisfies the double circuit property, thus the min-max formula of Dress and Lovász on the maximum matroid matching holds forM. 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 2-dimensional rigidity matroid (k = 2, l = 3).