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Let H be a (k + 1)-uniform, D-regular hypergraph on n vertices and U(H) be the minimum number of vertices left uncovered by a matching in H. Cj(H), the j-codegree of H, is the maximum number of edges sharing a set of j vertices in common. We prove a general upper bound on U(H), based on the codegree sequence C2(H), C3(H) . . . . Our bound improves and generalizes many results on the topic, including those of Grable [Gra], AlonKim-Spencer [AKS], and Kostochka-Rödl [KR]. It also leads to a substantial improvement in several applications. The key ingredient of the proof is the so-called polynomial technique, which is a new and useful tool to prove concentration results for functions with large Lipschitz coefficient. This technique is of independent interest. §

New bounds on nearly perfect matchings in hypergraphs: Higher codegrees do help