Exact minimum degree thresholds for perfect matchings in uniform hypergraphs

TL;DR

This paper establishes exact minimum degree thresholds for perfect matchings in certain uniform hypergraphs, improving previous asymptotic results by employing advanced combinatorial methods.

Contribution

It provides the precise minimum r-degree condition for perfect matchings in k-uniform hypergraphs when 4 divides k and k/2  r  k-1, enhancing prior asymptotic findings.

Findings

Derived exact degree thresholds for perfect matchings

Improved upon previous asymptotic results

Utilized advanced combinatorial techniques including the absorbing method

Abstract

Given positive integers k and r where 4 divides k and k/2 \leq r \leq k-1, we give a minimum r-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an asymptotically exact result. Our approach makes use of the absorbing method, as well as the hypergraph removal lemma and a structural result of Keevash and Sudakov relating to the Turan number of the expanded triangle.