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

Abstract

Let H be a (k+1)-uniform, D-regular hypergraph on n vertices and let (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 (H), based on the codegree sequence C2 (H), C3 (H),…. Our bound improves and generalizes many results on the topic, including those of Grable, Alon, Kim, and Spencer, and Kostochka and Rodl. 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. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 29–63, 2000