A note on perfect matchings in uniform hypergraphs

TL;DR

This paper determines exact minimum degree thresholds for perfect matchings in uniform hypergraphs, especially when fractional matchings are significantly easier to achieve, extending previous results and providing new specific thresholds.

Contribution

It extends prior work by establishing exact thresholds for all f6 and f6, and introduces two new specific thresholds for hypergraphs.

Findings

Exact minimum f6 thresholds for perfect matchings in hypergraphs.

New thresholds for (k, f6) = (5,2) and (7,3).

Comparison between thresholds for perfect and fractional matchings.

Abstract

We determine the \emph{exact} minimum $\ell$-degree threshold for perfect matchings in $k$-uniform hypergraphs when the corresponding threshold for perfect fractional matchings is significantly less than $\frac{1}{2} \binom{n}{k- \ell}$. This extends our previous results that determine the minimum $\ell$-degree thresholds for perfect matchings in $k$-uniform hypergraphs for all $\ell\ge k/2$ and provides two new (exact) thresholds: $(k,\ell)=(5,2)$ and $(7,3)$.