NEARLY PERFECT MATCHINGS IN UNIFORM HYPERGRAPHS

摘要

We prove that, for any integers k, l with k = 3 and k/2 l = k-1, there exists a positive real mu such that, for all sufficiently large integers m, n satisfying n/k -mu n = m = n/k - 1 - (1 - l/k) [k- l/2l- k], if H is a k-uniform hypergraph on n vertices and delta(l)(H) (n-1 k-1) - ((n -1)-m k-1), then H has a matching of size m + 1. This improves upon an earlier result of Han, Person, and Schacht for the range k/2 l = k -1. In many cases, our result gives a tight bound on delta(l)(H) for near perfect matchings (e.g., when l = geq 2k/3, n = r (mod k), 0 = r k, and r + l = k, we can take m = [ n/k]-2). When k = 3, using an absorbing lemma of Han, Person, and Schacht, our proof also implies a result of Kuhn, Osthus, and Treglown (and, independently, of Khan) on perfect matchings in 3-uniform hypergraphs.