Uniformity-independent minimum degree conditions for perfect matchings in hypergraphs

Abstract

In this note, we prove that there exists a universal constant $c=\frac{43}{50}$ such that for every $k\in \mathbb{N}$ and every $d<k/2$, every $k$-uniform hypergraph on $n$ vertices and with minimum $d$-degree at least $(c+o_n(1))\left(\genfrac{}{}{0ex}{}{n-d}{k-d}\right)$ contains a perfect matching. This is the first such bound which is independent of $k$, and therefore, improves all previously known bounds when $k$ is large. Our approach is based on combining the seminal work of Alon et al. with known bounds on a conjectured probabilistic inequality due to Feige.

After this note appeared on the arXiv, we were informed that similar observations are implicit in the literature; see Remark 1.6

References in corpus (1)

• Fractional and integer matchings in uniform hypergraphs

Cited by in corpus (1)

• Bounding probability of small deviation on sum of independent random variables: Combination of moment approach and Berry-Esseen theorem