Exact minimum degree thresholds for perfect matchings in uniform hypergraphs II

TL;DR

This paper establishes exact minimum r-degree conditions that guarantee perfect matchings in k-uniform hypergraphs, improving previous asymptotic results and extending the range of parameters for which these conditions are known.

Contribution

It provides the exact minimum degree thresholds for perfect matchings in uniform hypergraphs for a broader range of parameters, building on and refining prior asymptotic results.

Findings

Established exact minimum r-degree thresholds for perfect matchings

Extended previous results to a wider parameter range

Used the absorbing method to prove the thresholds

Abstract

Given positive integers k\geq 3 and r where 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, and builds on work in 'Exact minimum degree thresholds for perfect matchings in uniform hypergraphs', where we proved the result for k divisible by 4.