Improved Bound on Vertex Degree Version of Erd\H{o}s Matching Conjecture

For a $k$-uniform hypergraph $H$, let ${\delta }_1(H)$ denote the minimum vertex degree of $H$, and $\nu (H)$ denote the size of the largest matching in $H$. In this paper, we show that for any $k\ge 3$ and $\beta >0$, there exists an integer $n_0(\beta ,k)$ such that for positive integers $n\ge n_0$ and $m\leq (\frac{k}{2(k-1)}-\beta)\frac{n}{k}$,if$H$isan$n$-vertex$k$-graphwith$\delta_1(H)>{{n-1}\choose {k-1}}-{{n-m}\choose {k-1}},$then$\nu(H)\geq m$. This improves upon earlier results of Bollob'{a}s, Daykin and Erd\H{o}s (1976) for the range $n>2k^3(m+1)$ and Huang and Zhao (2017) for the range $n\geq 3k^2 m$.