On Perfect Matchings in Uniform Hypergraphs with Large Minimum Vertex Degree

We study sufficient l-degree (1 ≤ l < k) conditions for the appearance of perfect and nearly perfect matchings in k-uniform hypergraphs. In particular, we obtain a minimum vertex degree condition (l = 1) for 3-uniform hypergraphs, which is approximately tight, by showing that every 3-uniform hypergraph on n vertices with minimum vertex degree at least (5/9+o(1)) ` n 2 ́ contains a perfect matching. 1. Notations and Results Our notation follows [2]. We refer to the set {1, 2, . . . , n} with n ∈ N by [n]. For a set M and an integer k, we denote by ( M k ) = {A ⊆ M : |A| = k} the set of all k-element subsets of M and we denote by (M)k = {(v1, v2, . . . , vk) : {v1, . . . , vk} ∈