Uniformity-independent minimum degree conditions for perfect matchings in hypergraphs

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 $d<k\mathrm{/}2$, every $k$-uniform hypergraph on $n$ vertices and with minimum $d$-degree at least $(c+o_n(1))\left(\genfrac{}{}{0px}{}{n-d}{k-d}\right)$ 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.