A DEGREE SEQUENCE STRENGTHENING OF THE VERTEX DEGREE THRESHOLD FOR A PERFECT MATCHING IN 3-UNIFORM HYPERGRAPHS*

The study of asymptotic minimum degree thresholds that force matchings and tilings in hypergraphs is a lively area of research in combinatorics. A key breakthrough in this area was a result of Han, Person, and Schacht [SIAM J. Disc. Math., 23 (2009), pp. 732-748] who proved that the asymptotic minimum vertex degree threshold for a perfect matching in an n-vertex 3-graph is (5/9 + o(1) (n 2). In this paper, we improve on this result, giving a family of degree sequence results, all of which imply the result of H`an, Person and Schacht and additionally allow one-third of the vertices to have degree 1/9 (n 2) below this threshold. Furthermore, we show that this result is, in some sense, tight.