A degree sequence strengthening of the vertex degree threshold for a perfect matching in 3-uniform hypergraphs

Abstract

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 Hàn, Person and Schacht who proved that the asymptotic minimum vertex degree threshold for a perfect matching in an $n$-vertex $3$-graph is $\left(\frac{5}{9}+o(1)\right)\left(\genfrac{}{}{0ex}{}{n}{2}\right)$. In this paper we improve on this result, giving a family of degree sequence results, all of which imply the result of Hàn, Person and Schacht, and additionally allow one third of the vertices to have degree $\frac{1}{9}\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ below this threshold. Furthermore, we show that this result is, in some sense, tight.

arXiv comment24 pages (including appendix)