Fractional and integer matchings in uniform hypergraphs

TL;DR

This paper improves bounds on minimum degree conditions that guarantee perfect or large matchings in uniform hypergraphs, using fractional matching techniques to derive new asymptotically tight results.

Contribution

It introduces fractional matching bounds and translates them into improved integer matching bounds in uniform hypergraphs, extending previous results.

Findings

Improved bounds on minimum d-degree for perfect matchings.

Asymptotically tight bounds for minimum vertex degree for large matchings.

Extension of classical results to fractional and asymptotic regimes.

Abstract

Our main result improves bounds of Markstrom and Rucinski on the minimum d-degree which forces a perfect matching in a k-uniform hypergraph on n vertices. We also extend bounds of Bollobas, Daykin and Erdos by asymptotically determining the minimum vertex degree which forces a matching of size t < n/2(k-1) in a k-uniform hypergraph on n vertices. Further asymptotically tight results on d-degrees which force large matchings are also obtained. Our approach is to prove fractional versions of the above results and then translate these into integer versions.